Mathematical Analysis I PDF

Mathematical Analysis I Claudio Canute, Anita Tabacco Mathematical Analysis I ^ S p r iinger CLAUDIO CANUTO Dipartimento di Matematica Politecnico di Torino ANITA TABACCO Dipartimento di Matematica Politecnico di Torino Translated by: Simon G. Chiossi Institut fiir Mathematik Humboldt Univers...

Mathematical Analysis I Claudio Canute, Anita Tabacco Mathematical Analysis I ^ S p r iinger CLAUDIO CANUTO Dipartimento di Matematica Politecnico di Torino ANITA TABACCO Dipartimento di Matematica Politecnico di Torino Translated by: Simon G. Chiossi Institut fiir Mathematik Humboldt Universitat zu Berlin CIP-Code: 2008932358 ISBN 978-88-470-0875-5 Springer Milan Berlin Heidelberg New York e-ISBN 978-88-470-0876-2 Springer Milan Berlin Heidelberg New York Springer-Verlag is a part of Springer Science-HBusiness Media springer.com © Springer-Verlag Italia, Milan 2008 Printed in Italy This work is subject to copyright. All rights are reserved, whether the whole or part of the material is con- cerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, re- production on microfilm or in other ways, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the Italian Copyright Law. Files and figures produced by the Authors Cover-Design: Francesca Tonon, Milan Printing and Binding: Signum Sri, BoUate (MI) Springer-Verlag ItaHa Sri, Via Decembrio 28,20137 Milan-I Preface The recent European Programme Specifications have forced a reassessment of the structure and syllabi of the entire system of Italian higher education, and an ensuing rethinking of the teaching material. Nowadays many lecture courses, especially rudimentary ones, demand that stu- dents master a large amount of theoretical and practical knowledge in a span of just few weeks, in order to gain a small number of credits. As a result, instructors face the dilemma of how to present the subject matter. They must make appro- priate choices about lecture content, the comprehension level required from the recipients, and which kind of language to use. This textbook is meant to help students acquire the basics of Calculus in cur- ricula where mathematical tools play a crucial part (so Engineering, Physics, Com- puter Science and the like). The fundamental concepts and methods of Differential and Integral Calculus for functions of one real variable are presented with the pri- mary purpose of letting students assimilate their effective employment, but with critical awareness. The general philosophy inspiring our approach has been to sim- plify the system of notions available prior to the university reform; at the same time we wished to maintain the rigorous exposition and avoid the trap of compiling a mere formulary of ready-to-use prescriptions. From this point of view, the treatise is 'stratified' in three layers, each corre- sponding to increasingly deeper engagement by the user. The intermediate level corresponds to the unabridged text. Notions are first presented in a naive manner, and only later defined precisely. Their features are discussed, and computational techniques related to them are exhaustively explained. Besides this, the fundamen- tal theorems and properties are followed by proofs, which are easily recognisable by the font's colour. At the elementary level the proofs and the various remarks should be skipped. For the reader's sake, essential formulas, and also those judged important, have been highlighted in blue, and gray, respectively. Some tables, placed both through- out and at the end of the book, collect the most useful formulas. It was not our desire to create a hierachy-of-sorts for theorems, instead to leave the instructor free to make up his or her own mind in this respect. VI Preface The deepest-reaching level relates to an internet website, and enables the strongly motivated reader to explore further into the subject. We believe that the general objectives of the Programme Specifications are in line with the fact that willing and able pupils will build a solid knowledge, in the tradition of Italy's academic education. The book contains several links to a webpage where the reader will find complements to, and insight in various topics. In this fashion every result that is stated possesses a corresponding proof. To make the approach to the subject less harsh, and all the more gratifying, we have chosen a casual presentation in the first two chapters, where relevant definitions and properties are typically part of the text. From the third chapter onwards they are highlighted by the layout more discernibly. Some definitions and theorems are intentionally not stated in the most general form, so to privilege a brisk understanding. For this reason a wealth of examples are routinely added along the way right after statements, and the same is true for computational techniques. Several remarks enhance the presentation by underlining, in particular, special cases and exceptions. Each chapter ends with a large number of exercises that allow one to test on the spot how solid one's knowledge is. Exercises are grouped according to the chapter's major themes and presented in increasing order of difficulty. All problems are solved, and at least half of them chaperone the reader to the solution. We have adopted the following graphical conventions for the constituent build- ing blocks: definitions appear on a gray background, theorems' statements on blue. a vertical coloured line marks examples, and boxed exercises, like | 12. |, indicate that the complete solution is provided. An Italian version of this book has circulated for a number of years in Italy, and has been extensively tested at the Politecnico in Turin. We wish to dedicate this volume to Professor Guido Weiss of Washington University in St. Louis, a master in the art of teaching. Generations of students worldwide have benefited from Guido's own work as a mathematician; we hope that his own clarity is at least partly reflected in this textbook. We are thankful to the many colleagues and students whose advice, suggestions and observations have allowed us to improve the exposition. Special thanks are due to Dr. Simon Chiossi, for the careful and effective work of translation. Torino, June 2008 Claudio Canuto, Anita Tabacco Online additional material At the address http://calvino.polito.it/canuto-tabacco/analysis_l further material and complementary theory are available. These online notes are organised under the following headings: Principle of Mathematical Induction The number e Elementary functions Limits Continuous functions Sequences Numerical series Derivatives De THopital's Theorem Convex functions Taylor expansions Cauchy integral Riemann integral Improper integrals and provides, in particular, rigorous proofs for the statements that are not shown in the text. The reader is encouraged to refer to the relevant section whenever the symbol --^ appears throughout the treatise, as in ' ^ The nxmbex e. Contents Basic notions 1 1.1 Sets 1 1.2 Elements of mathematical logic 5 1.2.1 Connectives 5 1.2.2 Predicates 6 1.2.3 Quantifiers 7 1.3 Sets of numbers 8 1.3.1 The ordering of real numbers 12 1.3.2 Completeness of E 17 1.4 Factorials and binomial coefficients 18 1.5 Cartesian product 21 1.6 Relations in the plane 23 1.7 Exercises 25 1.7.1 Solutions 26 Functions 31 2.1 Definitions and first examples 31 2.2 Range and pre-image 36 2.3 Surjective and injective functions; inverse function 38 2.4 Monotone functions 41 2.5 Composition of functions 43 2.5.1 Translations, rescaUngs, reflections 45 2.6 Elementary functions and properties 47 2.6.1 Powers 48 2.6.2 Polynomial and rational functions 50 2.6.3 Exponential and logarithmic functions 50 2.6.4 Trigonometric functions and inverses 51 2.7 Exercises 56 2.7.1 Solutions 58 X Contents 3 Limits and continuity I 65 3.1 Neighbourhoods 65 3.2 Limit of a sequence 66 3.3 Limits of functions; continuity 72 3.3.1 Limits at infinity 72 3.3.2 Continuity. Limits at real points 74 3.3.3 One-sided Umits; points of discontinuity 82 3.3.4 Limits of monotone functions 85 3.4 Exercises 86 3.4.1 Solutions 86 4 Limits and continuity II 89 4.1 Theorems on limits 89 4.1.1 Uniqueness and sign of the limit 89 4.1.2 Comparison theorems 91 4.1.3 Algebra of limits. Indeterminate forms of algebraic type.... 96 4.1.4 Substitution theorem 102 4.2 More fundamental limits. Indeterminate forms of exponential type. 105 4.3 Global features of continuous maps 108 4.4 Exercises 115 4.4.1 Solutions 117 5 Local comparison of functions. Numerical sequences and series 123 5.1 Landau symbols 123 5.2 Infinitesimal and infinite functions 130 5.3 Asymptotes 135 5.4 Further properties of sequences 137 5.5 Numerical series 141 5.5.1 Positive-term series 146 5.5.2 Alternating series 149 5.6 Exercises 151 5.6.1 Solutions 154 6 Differential calculus 167 6.1 The derivative 167 6.2 Derivatives of the elementary functions. Rules of differentiation.... 170 6.3 Where differentiability fails 175 6.4 Extrema and critical points 178 6.5 Theorems of Rolle and of the Mean Value 181 6.6 First and second finite increment formulas 183 6.7 Monotone maps 185 6.8 Higher-order derivatives 187 6.9 Convexity and inflection points 189 6.9.1 Extension of the notion of convexity 192 6.10 Qualitative study of a function 193  Contents XI 6.10.1 Hyperbolic functions 195 6.11 The Theorem of de I'Hopital 197 6.11.1 Apphcations of de THopital's theorem 199 6.12 Exercises 201 6.12.1 Solutions 205 Taylor expansions and applications 223 7.1 Taylor formulas 223 7.2 Expanding the elementary functions 227 7.3 Operations on Taylor expansions 234 7.4 Local behaviour of a map via its Taylor expansion 242 7.5 Exercises 246 7.5.1 Solutions 248 Geometry in the plane and in space 257 8.1 Polar, cylindrical, and spherical coordinates 257 8.2 Vectors in the plane and in space 260 8.2.1 Position vectors 260 8.2.2 Norm and scalar product 263 8.2.3 General vectors 268 8.3 Complex numbers 269 8.3.1 Algebraic operations 270 8.3.2 Cartesian coordinates 271 8.3.3 Trigonometric and exponential form 273 8.3.4 Powers and nth roots 275 8.3.5 Algebraic equations 277 8.4 Curves in the plane and in space 279 8.5 Functions of several variables 284 8.5.1 Continuity 284 8.5.2 Partial derivatives and gradient 286 8.6 Exercises 289 8.6.1 Solutions 292 Integral calculus I 299 9.1 Primitive functions and indefinite integrals. 300 9.2 Rules of indefinite integration 304 9.2.1 Integrating rational maps 310 9.3 Definite integrals 317 9.4 The Cauchy integral 318 9.5 The Riemann integral 320 9.6 Properties of definite integrals 326 9.7 Integral mean value 328 9.8 The Fundamental Theorem of integral calculus 331 9.9 Rules of definite integration 335 9.9.1 Application: computation of areas 337 XII Contents 9.10 Exercises 340 9.10.1 Solutions 342 10 Integral calculus II 355 10.1 Improper integrals 355 10.1.1 Unbounded domains of integration 355 10.1.2 Unbounded integrands 363 10.2 More improper integrals 367 10.3 Integrals along curves 368 10.3.1 Length of a curve and arc length 373 10.4 Integral vector calculus 376 10.5 Exercises 378 10.5.1 Solutions 380 11 Ordinary differential equations 387 11.1 General definitions 387 11.2 First order differential equations 388 11.2.1 Equations with separable variables 392 11.2.2 Linear equations 394 11.2.3 Homogeneous equations 397 11.2.4 Second order equations reducible to first order 398 11.3 Initial value problems for equations of the first order 399 11.3.1 Lipschitz functions 399 11.3.2 A criterion for solving initial value problems 402 11.4 Linear second order equations with constant coefficients 404 11.5 Exercises 410 11.5.1 Solutions 412 Tables and Formulas 423 Index 429 Basic notions In this introductory chapter some mathematical notions are presented rapidly, which lie at the heart of the study of Mathematical Analysis. Most should already be known to the reader, perhaps in a more thorough form than in the following presentation. Other concepts may be completely new, instead. The treatise aims at fixing much of the notation and mathematical symbols frequently used in the sequel. 1.1 Sets We shall denote sets mainly by upper case letters X, F ,... , while for the members or elements of a set lower case letters x, ^ ,... will be used. When an element x is in the set X one writes x G X ('x is an element of X', or 'the element x belongs to the set X'), otherwise the symbol x 0 X is used. The majority of sets we shall consider are built starting from sets of numbers. Due to their importance, the main sets of numbers deserve special symbols, namely: N= set of natural numbers Z = set of integer numbers Q = set of rational numbers M= set of real numbers C = set of complex numbers. The definition and main properties of these sets, apart from the last one, will be briefiy recalled in Sect. 1.3. Complex numbers will be dealt with separately in Sect. 8.3. Let us fix a non-empty set X, considered as ambient set. A subset A of X is a set all of whose elements belong to X; one writes A C X ('A is contained, or included, in X') if the subset A is allowed to possibly coincide with X, and A C X {'A is properly contained in X') in case A is a proper subset of X, that  1 Basic notions Figure 1.1. Venn diagrams (left) and complement (right) is, if it does not exhaust the whole X. From the intuitive point of view it may be useful to represent subsets as bounded regions in the plane using the so-called Venn diagrams (see Fig. 1.1, left). A subset can be described by listing the elements of X which belong to it ^ = |a:,y,...,2^}; the order in which elements appear is not essential. This clearly restricts the use of such notation to subsets with few elements. More often the notation A=^{xeX \ p{x)} or A = {x eX : p{x)} will be used (read 'A is the subset of elements x of X such that the condition p{x) holds'); p{x) denotes the characteristic property of the elements of the subset, i.e., the condition that is valid for the elements of the subset only, and not for other elements. For example, the subset A of natural numbers smaller or equal than 4 may be denoted A-{0,1,2,3,4} or A = {xeN\x 2 holds, for p would not be the least of all upper bounds. Thus necessarily p^ — 2. Note that B, albeit contained in Q, is not allowed to have a rational upper bound, because p^ — 2 prevents p from being rational (Property 1.1). This example explains why the completeness of R lies at the core of the pos- sibility to solve in R many remarkable equations. We are thinking in particular about the family of algebraic equations x^ = a, (1.8) where n G N+ and a G R, for which it is worth recalling the following known fact. Property 1,8 i) Let n € N^. be odd. Then for any a € E equation (1.8) has exactly one solution in K, denoted by x = ^ orx~ a^^'^ and called the nth root of a, a) Let n € N4- be even. For any a > 0 equation (1.8) has two real solutions wiih the same absolute value but opposite signs; when a = 0 there is one solution X = 0 only; for a < 0 there are no solutions in R. The non-negative solution is indicated by a: =^ \/aorx = a^/^, and called the nth (arithmetic) root of a. 1.4 Factorials and binomial coefficients We introduce now some noteworthy integers that play a role in many areas of Mathematics. Given a natural number n > 1, the product of all natural numbers between 1 and n goes under the name of factorial of n and is indicated by n\ (read 'n factorial'). Out of conveniency one sets 0! = 1. Thus 0! = 1, 1! = 1, n! = l - 2.... - n = ( n ~ - l ) ! n for n > 2. (1.9) Factorials grow extremely rapidly as n increases; for instance 5! = 120, 10! = 3628800 and 100! > 10^^^ E x a m p l e 1.9 Suppose we have n > 2 balls of different colours in a box. In how many ways ( can we extract the balls from the box?  1.4 Factorials and binomial coefficients 19 When taking the first ball we are making a choice among the n balls in the box; the second ball will be chosen among the n - 1 balls left, the third one among n — 2 and so on. Altogether we have n{n — 1)... 2 1 = n! different ways to extract the balls: n! represents the number of arrangements of n distinct objects in a sequence, called permutations of n ordered objects. If we stop after k extractions, 0 < fc < n, we end up with n{n — 1 )... (n — /c + 1) n\ possible outcomes. The latter expression, also written as ——r, is the number (n — k)l of possible permutations of n distinct objects in sequences of k objects. If we allow repeated colours, for instance by reintroducing in the box a ball of the same colour as the one just extracted, each time we choose among n. After k > 0 choices there are then n^ possible sequences of colours: n^ is the number of permutations of n objects in sequences of k, with repetitions (i.e., allowing an object to be chosen more than once). D Given two natural numbers n and k such that 0 < A: < n, one calls binomial coefficient the number ni (1.10) k\{n - fc)! (the symbol (^) is usually read 'n choose fc'). Notice that if 0 < fc < n n! = l-...-n = l-...-(n-fc)(n-fc+l)-...-(71-1)71 = (n-fc)!(n-fc+l)-...-(n-l)?!, so simplifying and rearranging the order of factors at the numerator, (1.10) be- comes (n\ n{n — 1)... ( n - f c + l) (1.11) \\kj- fc! another common expression for the binomial coefficient. From definition (1.10) it follows directly that ^n\ f n ,k) \n - fc; and = 1, n n. n-l Moreover, it is easy to prove that for any n > 1 and any fc with 0 < fc < n n-l n-l (1.12) fc-1 fc which provides a convenient means for computing binomial coefficients recursively^ the coefficients relative to n objects are easily determined once those involving n — 1 objects are computed. The same formula suggests to write down binomial 20 1 Basic notions coefficients in a triangular pattern, known as PascaVs triangle^ (Fig. 1.7): each coefficient of a given row, except for the I's on the boundary, is the sum of the two numbers that he above it in the preceding row, precisely as (1.12) prescribes. The construction of Pascal's triangle shows that the binomial coefficients are natural numbers. 1 11 1 2 1 1 3 3 1 1 4 6 4 1 1...... 1 Figure 1.7. Pascal's triangle The term 'binomial coefficient' originates from the power expansion of the polynomial a -\-h m terms of powers of a and h. The reader will remember the important identities (a -h hf = a^ + 2ab + h^ and (a + bf = a^ + ^aH + 3a6^ + b^. The coefficients showing up are precisely the binomial coefficients for n = 2 and n = 3. In general, for any n > 0, the formula (a + 6)" = a" + naJ"-'h+...+ la"--kf^k +.. + nab"-- "^+6" n k=0 ^^yn^ \i\,y (1.13) holds, known as (Newton's) binomial expansion. This formula is proven with (1.12) using a proof by induction ^^ Principle of Mathematical Induction. Example 1.9 (continuation) Given n balls of different colours, let us fix k with 0 < A: < n. How many different sets of k balls can we form? Extracting one ball at a time for k times, we already know that there are n{n — 1 )... (n —fc+ 1) outcomes. On the other hand the same k balls, extracted in a different order, wiU yield the same set. Since the possible orderings of k elements are fc!, we see that the number of distinct sets of k balls chosen from n is -^, = ( 7 I This coefficient represents the number of A:! \kj combinations of n objects taken k at a. time. Equivalently, the number of subsets of k elements of a set of cardinality n. ^ Sometimes the denomination Tartaglia's triangle appears.  1.5 Cartesian product 21 Formula (1.13) with a = b = 1 shows that the sum of ah binomial coefficients with n fixed equals 2^, non-incidentally also the total number of subsets of a set with n elements. 1.5 Cartesian product Let X, y be non-empty sets. Given elements x in X and y in F , we construct the ordered pair of numbers whose first component is x and second component is y. An ordered pair is concep- tually other than a set of two elements. As the name says, in an ordered pair the order of the components is paramount. This is not the case for a set. li x y^ y the ordered pairs {x,y) and (y^x) are distinct, while {x^y} and {y,x} coincide as sets. The set of all ordered pairs (x^y) when x varies in X and y varies in Y is the Cartesian product of X and F , which is indicated hy X xY. Mathematically, XxY = {{x,y)\xeX, y€Y}. The Cartesian product is represented using a rectangle, whose basis corre- sponds to the set X and whose height is Y (as in Fig. 1.8). If the sets X, Y are different, the product X xY will not be equal to F x X, in other words the Cartesian product is not commutative. But if y — X, it is customary to put X x X = X^ for brevity. In this case the subset of X^ A = {{x,y)eX^ \x = y} of pairs with equal components is called the diagonal of the Cartesian product. Figure 1.8. Cartesian product of sets 22 1 Basic notions The most significant example of Cartesian product stems from X = Y = 'R. The set R^ consists of ordered pairs of real numbers. Just as the set R mathematically represents a straight line, so R^ is a model of the plane (Fig. 1.9, left). In order to define this correspondence, choose a straight line in the plane and fix on it an origin O, a positive direction and a length unit. This shall be the x-axis. Rotating this line counter-clockwise around the origin by 90° generates the y-axis. In this way we have now an orthonormal frame (we only mention that it is sometimes useful to consider frames whose axes are not orthogonal, and/or the units on the axes are different). Given any point P on the plane, let us draw the straight lines parallel to the axes passing through the point. Denote by x the real number corresponding to the intersection of the x-axis with the parallel to the y-axis, and by y the real number corresponding to the intersection of the y-axis with the parallel to the x-axis. An ordered pair (x, y) G R^ is thus associated to each point P on the plane, and vice versa. The components of the pair are called (Cartesian) coordinates of P in the chosen frame. The notion of Cartesian product can be generalised to the product of more sets. Given n non-empty sets Xi, X 2 ,... , X^, one considers ordered n—tuples (xi,X2,...,Xn) where, for every i = 1,2,..., n, each component Xi lives in the set Xi. The Carte- sian product Xi X X2 X... X Xn is then the set of all such n—tuples. When Xi = X2 =.. = Xn = X one simply writes X x X x... x X = X^. In particular, R^ is the set of triples [x^y^z] of real numbers, and represents a mathematical model of three-dimensional space (Fig. 1.9, right). Figure 1.9. Models of the plane (left) and of space (right)  1.6 Relations in the plane 23 1.6 Relations in the plane We call Cartesian plane a plane equipped with an orthonormal Cartesian frame built as above, which we saw can be identified with the product R^. Every non-empty subset R of R^ defines a relation between real numbers; precisely, one says x is R-related to y^ or x is related to y by i?, if the ordered pair (x, y) belongs to R. The graph of the relation is the set of points in the plane whose coordinates belong to R. A relation is commonly defined by one or more (in)equalities involving the variables x and y. The subset R is then defined as the set of pairs (x, y) such that X and y satisfy the constraints. Finding R often means determining its graph in the plane. Let us see some examples. Examples 1.10 i) An equation like ax -\-by = c, with a, b constant and not both vanishing, defines a straight line. If 6 = 0, the line is parallel to the ^-axis, whereas a — 0 yields a parallel to the x-axis. Assuming 6 7^ 0 we can write the equation as y — mx + g, where m = — | and ^ = f The number m is called slope of the line. The line can be plotted by finding the coordinates of two points that belong to it, hence two distinct pairs (x, y) solving the equation. In particular c = 0 (or g' = 0) if and only if the origin belongs to the line. The equation x — y = 0 for example defines the bisectrix of the first and third quadrants of the plane. ii) Replacing the ' = ' sign by ' < ' above, consider the inequality ax -^by < c. It defines one of the half-planes in which the straight line of equation ax-\-by = c divides the plane (Fig. 1.10). If 6 > 0 for instance, the half-plane below the line is obtained. This set is open, i.e., it does not contain the straight line, since the inequality is strict. The inequality ax ^by < c defines instead a closed set, i.e., including the line. Figure 1.10. Graph of the relation of Example 1.10 ii) 24 1 Basic notions iii) The system '?/>0, x-y>0, defines the intersection between the open half-plane above the x-axis and the closed half-plane lying below the bisectrix of the first and third quadrants. Thus the system describes (Fig. 1.11, left) the wedge between the positive x-axis and the bisectrix (the points on the x-axis are excluded). iv) The inequality \x-y\o Xy/jx^ - 4 | _ ^ ) ^1x2-41 -X >0 >0 x^ - 4 2. Describe the following subsets of M; [a^l A = {x G M : x^ + 4x + 13 < 0} n {x G M : 3x2 + 5 > 0} 3x + l b) B = {x G R : (x + 2)(x - l)(x - 5) < 0} n {x G M : >0} x-2 x^ - 5x 4- 4 c) C = {xe < 0} U {x G M : \/7x + l + X = 17} x2-9 d) L) = { X G R : X - 4 > V:^2 _ g^ _^ 5} U { X G M : X + 2 > \ / ^ " ^ } 26 1 Basic notions 3. Determine and draw a picture of the following subsets ofM!^: [^ A = {{x,y) eR'^ :xy>0} h) B = {{x,y) G E^ : x^ - y^ > o} c) C = {{x,y)eR^:\y-x^\ 1} 4 e) E = {{x,y)eR^ :l + xy>0} f) F = {{x,y) eR^ : x-y y^ 0} 4. Tell whether the following subsets ofR are bounded from above and/or below, specifying upper and lower bounds, plus maximum and minimum (if existent): a) {x eR: x = n 01 X =^ —, neN\ {0}} n^ b) B = {x eR : -1< X < 1 OT X = 20} 2n-3 c) C = { X G M : 0 < X < 1 o r x = - , n G N \ { 0 , 1}} —I n—1 d) D = {zeR: z = xy with x, t/ G M, - 1 < x < 2, - 3 < y < - 1 } 1.7.1 Solutions 1. Inequalities: a) This is a fractional inequality. A fraction is positive if and only if numerator and denominator have the same sign. As N{x) = 2x — l > O i f x > 1/2, and D{x) = X — 3 > 0 for X > 3, the inequality holds when x < 1/2 or x > 3. b) - | < x < f c) Shift all terms to the left and simplify: X- 1 2x - 3 -x2 + 3x - 3 >0, I.e., >0. x-2 x-3 (x-2)(x-3) The roots of the numerator are not real, so N{x) < 0 always. The inequality thus holds when D{x) < 0, hence 2 < x < 3. d) Moving terms to one side and simplifying yields: x+1 |x|(2a;-l)-a;^ + l >0, I.e. >0. X- 1 2x {x - l)(2a; - 1) Since |a;| = a; for x > 0 and |ar| = —x for a; < 0, we study the two cases separately. When X >0 the inequahty reads 2a;2 - X - x^ + 1 x+1 >0, or >0. (x - l)(2x - 1) (x - l)(2x - 1)  1.7 Exercises 27 The numerator has no real roots, hence x^ — x + 1 > 0 for all x. Therefore the inequality is satisfied if the denominator is positive. Taking the constrain X > 0 into account, this means 0 < x < 1/2 or x > 1. When X < 0 we have ~2x^ -h X - x^ -h 1 ^. -3x2 + X + 1 (x-l)(2x-l) > 0 ,' I.e., * '' ( x - l— ) ( 2 x - lr)T > 0. N{x) is annihilated by xi = ^ ~ ^ and X2 = ^"^/^, so 7V(x) > 0 for xi < x < X2 (notice that xi < 0 and X2 G (^,1))- As above the denominator is positive when X < 1/2 and x > 1. Keeping x < 0 in mind, we have Xi < x < 0. The initial inequality is therefore satisfied by any x G (xi, ^) U (1, -hoc). e) - 5 < X < - 2 , - | < X < 1, 1 < X < ^ ± ^ ; f) x < - §. g) First of all observe that the right-hand side is always > 0 where defined, hence when x^ — 2 x > 0 , i. e. , x < 0 o r x > 2. The inequality is certainly true if the left-hand side x — 3 is < 0, so for x < 3. If X — 3 > 0, we take squares to obtain 2 9 1 9 X — 6x + 9 < X — 2x , i.e., 4x > 9 , whence x>-. Gathering all information we conclude that the starting inequality holds wher- ever it is defined, that is for x < 0 and x > 2. h) X G [ - 3 , - A / 3 ) U ( V ^ , -hoc). i) As \x^ — 4| > 0, \/\x'^ — 4| is well defined. Let us write the inequality in the form Vk2-4| >x. If X < 0 the inequality is always true, for the left-hand side is positive. If x > 0 we square: \x^ - 4| > x ^ Note that 2 ,1 fx^ 1X^-4 ifx2, r-4| a + 4 if - 2- < X < 2. - I - -x^ Consider the case x > 2 first; the inequality becomes x^ — 4 > x"^, which is never true. Let now 0 < x < 2; then - x ^ -h 4 > x^, hence x^ - 2 < 0. Thus 0 < x < \/2 must hold. In conclusion, the inequality holds for x < A/2. ^) xG (-2,-A/2)U(2,+OC). 2. Subsets of R: a) Because x^ -h 4x -h 13 = 0 cannot be solved over the reals, the condition x^ -h 4x -h 13 < 0 is never satisfied and the first set is empty. On the other hand, 3x2 _^ 5 -> Q holds for every x G M, therefore the second set is the whole M. Thus ^ = 0 n R = 0. 28 1 Basic notions b) 5 = ( - o c , - 2 ) U ( 2 , 5 ). c) We can write x^ - 5 x + 4 _ ( x - 4 ) ( x - 1) ^2-9 ~ ( J : - 3 ) ( X + 3) ' whence the first set is (—3,1) U (3,4). To find the second set, let us solve the irrational equation \Jlx + 1 + x = 17, which we write as \/lx + 1 = 17—x. The radicand must necessarily be positive, hence x > — y. Moreover, a square root is always > 0, so we must impose 17 — X > 0, i.e., X < 17. Thus for —y < x < 17, squaring yields 7x + 1 = (17 - xf , x^ - 41x + 288 = 0. The latter equation has two solutions xi = 9, X2 = 32 (which fails the con- straint X < 17, and as such cannot be considered). The second set then contains only X = 9. Therefore C = (-3,1) U (3,4) U {9}. d) D=[1,+(X)). 3. Subsets of R^- a) The condition holds if x and y have equal signs, thus in the first and third quadrants including the axes (Fig. 1.13, left). b) See Fig. 1.13, right. c) We have y — x'^ if 2/ > ^^ , 1 ^ - ^ 1= 1 2 -r / 2 y It y < x^. Demanding y > x^ means looking at the region in the plane bounded from below by the parabola y — x^. There, we must have 1/ - x^ < 1, i.e., ?/ < x^ + 1, Figure 1.13. The sets A and B of Exercise 3  1.7 Exercises 29 Figure 1.14. The sets C and D of Exercise 3 that \s x^ x^ — \ ^ hence x^ — 1 < y < x^. Eventually, the required region is confined by (though does not include) the parabolas y = x'^ - I and y = x'^ + 1 (Fig. 1.14, left). d) See Fig. 1.14, right. e) For X > 0 the condition 1 -\- xy > 0 is the same as ?/ > —-. Thus we consider all points of the first and third quadrants above the hyperbola y = —-. 1 ^ For X < 0, l-\-xy > 0 means y < —-, satisfied by the points in the second and ^ 1 fourth quadrants this time, lying below the hyperbola y = —^^ At last, ifx = 0, 1 -h xy > 0 holds for any ?/, implying that the ^-axis belongs to the set E. Therefore: the region lies between the two branches of the hyperbola (these are not part of E") y = — ^, including the y-axis (Fig. 1.15, left). f) See Fig. 1.15, right. Figure 1.15. The sets E and F of Exercise 3 30 1 Basic notions 4. Bounded and unbounded sets: a) We have A = { 1 , 2 , 3 ,... , | , | , ^ ,... }. Since N \ {0} C A, the set A is not bounded from above, hence sup A — H-oo and there is no maximum. In addition, the fact that every element of A is positive makes A bounded from below. We claim that 0 is the greatest lower bound of A. In fact, if r > 0 were a lower bound of A, then ^ > r for any non-zero n G N. This is the same as n^ < ^, hence n < 4^. But the last inequality is absurd since natural numbers are not bounded from above. Finally 0 ^ A, so we conclude inf A = 0 and A has no minimum. b) ini B = —1, supB = maxB = 20, and minB does not exist. c) C = [0,1] U { | , | , | , | ,... } C [0,2); then C is bounded, and inf C = min C = 0. Since — =2 - , it is not hard to show that supC = 2, although n—1 n—1 there is no maximum in C. d) inf C = min C = —6, sup B = max B = 3. Functions Functions crop up regularly in everyday life (for instance: each student of the Polytechnic of Turin has a unique identification number), in physics (to each point of a region in space occupied by a fluid we may associate the velocity of the particle passing through that point at a given moment), in economy (each working day at Milan's stock exchange is tagged with the Mibtel index), and so on. The mathematical notion of a function subsumes all these situations. 2.1 Definitions and first examples Let X and Y be two sets. A function / defined on X with values in Y is a correspondence associating to each element x e X at most one element y E Y. This is often shortened to 'a function from X to Y\ A synonym for function is map. The set of a: G X to which / associates an element in Y is the domain of / ; the domain is a subset of X, indicated by d o m /. One writes /:dom/CX^y. If dom / = X, one says that / is defined on X and writes simply f : X ^ Y. The element y EY associated to an element x G dom / is called the image of X by or under / and denoted y = f{x). Sometimes one writes f :x^ f{x). The set of images y — f{x) of all points in the domain constitutes the range of / , a subset of Y indicated by i m /. The graph of / is the subset r ( / ) of the Cartesian product X xY made of pairs (x, f{x)) when x varies in the domain of / , i.e.. r{f) = {{xJ{x))eXxY : xedom/}. (2.1) 32 2 Functions Figure 2.1. Naive representation of a function using Venn diagrams In the sequel we shall consider maps between sets of numbers most of the time. If y == R, the function / is said real or real-valued. If X = R, the function is of one real variable. Therefore the graph of a real function is a subset of the Cartesian plane R^. A remarkable special case of map arises when X = N and the domain contains a set of the type {n G N : n > no} for a certain natural number no > 0. Such a function is called sequence. Usually, indicating by a the sequence, it is preferable to denote the image of the natural number n by the symbol a^ rather than a{n); thus we shall write a : n i-^ a^. A common way to denote sequences is {an}n>no (ignoring possible terms with n < no) or even {an}- Examples 2.1 Let us consider examples of real functions of real variable. i) / : R —> R, f{x) = ax-\-b {a,b real coefficients), whose graph is a straight line (Fig. 2.2, top left). ii) / : R ^^ R, f{x) = x^, whose graph is a parabola (Fig. 2.2, top right). iii) / : R\{0} C R -^ R, f{x) = ^, has a rectangular hyperbola in the coordinate system of its asymptotes as graph (Fig. 2.2, bottom left). iv) A real function of a real variable can be defined by multiple expressions on different intervals, in which case is it cahed a piecewise function. An example is given by / : [0,3] -^ R 3x ifO0, / : R - -Z, f{x) = sign(a;) = 1 0 ifa; = 0, l-l ifa; 0. The first few terms read 1 2 - 3 ao = 0, ai = - = 0.5, a2 = - = 0.6, as = -= 0.75. Its graph is shown in Fig. 2.4 (top left). x) The sequence an={l + iy (2.4) is defined for n > 1. The first terms are 9 64 625 ai = 2, a2 = - = 2.25, as =--= 2.37037, a^ =--= 2.44140625. 4 27 256 Fig. 2.4 (top right) shows the graph of such sequence.  2.1 Definitions and first examples 35 Figure 2.4. Clockwise: graphs of the sequences (2.3), (2.4), (2.6), (2.5) xi) The sequence an = n\ (2.5) associates to each natural number its factorial, defined in (1.9). The graph of this sequence is shown in Fig. 2.4 (bottom left); as the values of the sequence grow rapidly as n increases, we used different scalings on the coordinate axes. xii) The sequence f +1 if n is even, «" = (-!)" = 1-1 if n is odd, (-^0) (2-6) has alternating values +1 and —1, according to the parity of n. The graph of the sequence is shown in Fig. 2.4 (bottom right). At last, here are two maps defined on R^ (functions of two real variables). xiii) The function maps a generic point P of the plane with coordinates (x, y) to its distance from the origin. xiv) The map /:R2^M^ f{x,y) = {y,x) associates to a point P the point P' symmetric to P with respect to the bisectrix of the first and third quadrants. 36 2 Functions Consider a map from X to Y. One should take care in noting that the symbol for an element of X (to which one refers as the independent variable) and the symbol for an element in Y {dependent variable), are completely arbitary. What really determines the function is the way of associating each element of the domain to its corresponding image. For example, \i x^y^z^t are symbols for real numbers, the expressions y = f{x) = 3x, x = f{y) = 3?/, or z = f{t) = Zt denote the same function, namely the one mapping each real number to its triple. 2.2 Range and pre-image Let A be a subset of X. The image of A under / is the set f{A) = {f{x) : xeA}C{mf of all the images of elements of A. Notice that f{A) is empty if and only if A contains no elements of the domain of /. The image f{X) of the whole set X is the range of / , already denoted by i m /. Let y be any element of F ; the pre-image of ^ by / is the set r\y)^{x&domf : f{x)^y} of elements in X whose image is y. This set is empty precisely when y does not belong to the range of /. If 5 is a subset of F , the pre-image of B under / is defined as the set f-\B) = {xedomf : f{x)eB}, union of all pre-images of elements of B. It is easy to check that A C f~^{f{A)) for any subset A of d o m / , and f{f-^{B)) = Bnimf CBior any subset B of Y. Example 2.2 Let / : R -> M, f{x) = x^. The image under / of the interval A = [1, 2] is the interval B = [1,4]. Yet the pre-image of B under / is the union of the intervals [—2, —1] and [1,2], namely, the set f-\B) = {xeR : 1 < |x| < 2 } (see Fig. 2.5). The notions of infimum, supremum, maximum and minimum, introduced in Sect. 1.3.1, specialise in the case of images of functions.  2.2 Range and pre-image 37 y = fix) y = fix) 4 ^ i/(^) 15 1 i 1 2 -2 \ ^ 1 I 1 / 2 Figure 2.5. Image (left) and pre-image (right) of an interval relative to the function fix) = x' Definition 2.3 Let f be a real map and A a subset of domf. One calls supremum of / on A (or in A) the supremum of the image of A under f s u p / ( x ) = s u p / ( ^ ) = sup{/(x) \xe A), xeA Then f is bounded from above on A if the set f{A) is bounded from above, or equivalently, if sup f{x) < -hoc. xeA If sup f{x) is finite and belongs to fiA), then it is the maximum of this set. xeA This number is the maximum value (or simply, the maximum^ of / on A and is denoted by m a x / ( x ). x€A The concepts of infimum and of minimum of f on A are defined similarly. Eventually, f is said bounded on A if the set f{A) is bounded. At times, the shorthand notations supy^ / , maxyj^ / , et c. are used. The maximum value M = max^ / of / on the set A is characterised by the conditions: i) M is a value assumed by the function on A, i.e., there exists XM ^ A such that / ( X M ) = M] ii) M is greater or equal than any other value of the map on A, so for any xeA, fix) < M. E x a m p l e 2.4 Consider the function fix) defined in (2.2). One verifies easily max f(x) = 3, min f(x) — 0, max fix) — 3, inf fix) = 1. The map does not assume the value 1 anywhere in the interval [1,3], so there is no minimum on that set. 38 2 Functions 2.3 Surjective and injective functions; inverse function A map with values in Y is called onto if i m / = Y. This means that each y E Y is the image of one element x e X at least. The term surjective (on Y) has the same meaning. For instance, / : R -^ R, f{x)=ax-j-h with a 7^ 0 is surjective on R, or onto: the real number y is the image of x = ^ ^. On the contrary, the function / : R -^ R, f{x)=x'^is not onto, because its range coincides with the interval [0,+oo). A function / is called one-to-one (or 1-1) if every ^ G i m / is the image of a unique element x G d o m /. Otherwise put, if ^ = / ( ^ i ) = /(^2) for some elements xi,X2 in the domain of / , then necessarily xi = X2- This, in turn, is equivalent to Xi i-X2 f{xi)^f{x2) for all Xi,X2 G d o m / (see Fig. 2.6). Again, the term injective may be used. If a map / is one-to-one, we can associate to each element y in the range the unique x in the domain with f{x) = y. Such correspondence determines a function defined on Y and with values in X, called inverse function of / and denoted by the symbol f~^. Thus x = f~^{y) ^^ y = f{x) (the notation mixes up deliberately the pre-image of y under / with the unique element this set contains). The inverse function f~^ has the image of / as its domain, and the domain of / as range: dom / 1 = im / , im / " dom /. Figure 2.6. Representation of a one-to-one function and its inverse  2.3 Surjective and injective functions; inverse function 39 A one-to-one map is therefore invertible; the two notions (injectivity and invert- ibility) coincide. What is the link between the graphs of / , defined in (2.1), and of the inverse function /~^? One has nr') = {{yJ-Hy))^yxX -. yedomf-'} = {{f{x),x)eY xX : xedomf}. Therefore, the graph of the inverse map may be obtained from the graph of / by swapping the components in each pair. For real functions of one real variable, this corresponds to a reflection in the Cartesian plane with respect to the bisectrix y — X (see Fig. 2.7: a) is reflected into b)). On the other hand, finding the explicit expression x = f~^{y) of the inverse function could be hard, if possible at all. Provided that the inverse map in the form x — f~^{y) can be determined, often one prefers to denote the independent variable (of f~^) by x, and the dependent variable by ^, thus obtaining the expression y — f~^{x). This is merely a change of notation (see the remark at the end of Sect. 2.1). The procedure allows to draw the graph of the inverse function in the same frame system of / (see Fig. 2.7, from b) to c)). i , y y = X x = r\y) y^ --fix) dom/ im/ V'' a) dom/ X im/-^ cj I dom/ ^ X Figure 2.7. From the graph of a function to the graph of its inverse 40 2 Functions E x a m p l e s 2.5 i) The function / : R ^ R, f{x) = ax + bis one-to-one for all a 7^ 0 (in fact, / ( ^ i ) = /(^2) => CLXi = ax2 ^ Xi = X2). Its inverse is a: = f~^{y) = ^ ^ , or ii) The map / : R ^> R, f{x)=x'^is not one-to-one because f{x) = f{—x) for any real x. Yet if we consider only values > 0 for the independent variable, i.e., if we restrict / to the interval [0, +CXD), then the function becomes 1-1 (in fact, f{xi) = /(^2) =^ xl — X2 = {xi — a:2)(3:^i + X2) = 0 =^ xi = X2). The inverse function x = f~^{y) = y/y is also defined on [0,+oc). Conventionally one says that the 'squaring' map y = x'^ has the function 'square root' y = y/x for inverse (on [0, +00)). Notice that the restriction of / to the interval (—(X),0] is 1-1, too; the inverse in this case is y = —y/x. iii) The map / : R ^ R, /(x) = x^ is one-to-one. In fact f{x\) = /(X2) =^ x\— x\ = {X\ — X2){x\ + X\X2 + X2) = 0 ^ X\ = X2 siuCC x f + X\X2 ^ x\ — \\x\ + ^2 + (a^i + 3:2)^] > 0 for any x\ 7^ X2. The inverse function is the 'cubic root' y — ^ , defined on all R. As in Example ii) above, if a function / is not injective over the whole domain, it might be so on a subset A C dom /. The restriction of / to A is the function j\^\A^Y such that /|^ {x) = j[x), Vx G ^ , and is therefore invertible. Let / be defined on X with values F. If / is one-to-one and onto, it is called a bijection (or bijective function) from X to Y. If so, the inverse map j ~ ^ is defined on y , and is one-to-one and onto (on X); thus, f~^ is a bijection from Y toX. For example, the functions /(x) = ax + 6 (a ^ 0) and /(x) = x^ are bijections from R to itself. The function / ( x ) = x^ is a bijection on [0,+oc) (i.e., from [0,+oo) to [0,+(X))). If / is a bijection between X and F , the sets X and Y are in bijective cor- rispondence through / : each element of X is assigned to one and only one element of F , and vice versa. The reader should notice that two finite sets (i.e., containing a finite number of elements) are in bijective correspondence if and only if they have the same number of elements. On the contrary, an infinite set can correspond bijectively to a proper subset; the function (sequence) / : N ^ N, / ( n ) = 2n, for example, establishes a bijection between N and the subset of even numbers. To conclude the section, we would like to mention a significant interpretation of the notions of 1-1, onto, and bijective maps just introduced. Both in pure Math- ematics and in applications one is frequently interested in solving a problem, or an equation, of the form /(^) = y ,  2.4 Monotone functions 41 where / is a suitable function between two sets X and Y. The quantity y represents the datum of the problem, while x stands for the solution to the problem, or the unknown of the equation. For instance, given the real number y^ find the real number x solution of the algebraic equation Well, to say that / is an onto function on Y is the same as saying that the problem or equation of concern admits at least one solution for each given y mY] asking / to be 1-1 is equivalent to saying the solution, if it exists at all, is unique. Eventually, / bijection from X to F means that for any given y mY there is one, and only one, solution x G X. 2.4 Monotone functions Let / be a real map of one real variable, and / the domain of / or an interval contained in the domain. We would like to describe precisely the situation in which the dependent variable increases or decreases as the independent variable grows. Examples are the increase in the pressure of a gas inside a sealed container as we raise its temperature, or the decrease of the level of fuel in the tank as a car proceeds on a highway. We have the following definition. Definition 2.6 The function f is increasing on / if, given elements Xi, X2 in I with xi < X2, one has f{xi) < f{x2); in symbols Vxi ,^2 e ^, Xi < X2 => f{xi) < f{X2). (2.7) The function f is strictly increasing on / if Vxi X2 e / , Xi < X2 ^ f{xi) < f{x2). (2.8) y = f{x) y = f{x) f{xi) f{xi) = /(X2) ; / : Xi X2 I Xl X2 Figure 2.8. Strictly increasing (left) and decreasing (right) functions on an interval / 42 2 Functions If a map is strictly increasing then it is increasing as well, hence condition (2.8) is stronger than (2.7). The definitions of decreasing and strictly decreasing functions on / are obtained from the previous definitions by reverting the inequality between f{xi) and f{x2). The function / is (strictly) monotone on / if it is either (strictly) increasing or (strictly) decreasing on /. An interval / where / is monotone is said interval of monotonicity of /. Examples 2.7 i) The map / : R -^ R, f{x) = ax + 6, is strictly increasing on R for a > 0, constant on R for a = 0 (hence increasing as well as decreasing), and strictly decreasing on R when a < 0. ii) The map / : R -^ R, f{x) = x'^ is strictly increasing on 7 = [0, +oc). Taking in fact two arbitrary numbers xi,0^2 > 0 with xi < X2, we have x\ < X1X2 < x^. Similarly, / is strictly decreasing on (—oc,0]. It is not difficult to check that all functions of the type y = x'^, with n > 4 even, have the same monotonic behaviour as / (Fig. 2.9, left). iii) The function / : R ^ R, f{x)=x^ strictly increases on R. All functions like y = x'^ with n odd have analogous behaviour (Fig. 2.9, right). iv) Referring to Examples 2.1, the maps y = [x] and y = sign(a:) are increasing (though not strictly increasing) on R. The mantissa y = M{x) of x, instead, is not monotone on R; but it is nevertheless strictly increasing on each interval [ n , n - h l ) , n G Z. -1 -1 Figure 2.9. Graphs of some functions y = x^ with n even (left) and n odd (right)  2.5 Composition of functions 43 Now to a simple yet crucial result. Proposition 2.8 / / / is strictly monotone on its domain, then f is one-to- one. Proof. To fix ideas, let us suppose / is strictly increasing. Given Xi,X2 G d o m / with xi / X2, then either xi < X2 or X2 < xi. In the former case, using (2.8) we obtain / ( x i ) < /(X2), hence f{xi) ^ /(X2). In the latter case the same conclusion holds by swapping the roles of xi and X2. Under the assumption of the above proposition, there exists the inverse function f~^ then; one can comfortably check that f~^ is also strictly monotone, and in the same way as / (both are strictly increasing or strictly decreasing). For instance, the strictly increasing function / : [0, +00) -^ [0, +CXD), f{x) — x^ has, as inverse, the strictly increasing function f~^ : [0,+CXD) -^ [0,+00), f~^{x) = ^/x. The logic implication / is strictly monotone on its domain =4> / i s one-to-one cannot be reversed. In other words, a map / may be one-to-one without increasing strictly on its domain. For instance / : R ^ R defined by 1 if X 7^ 0, /(^) = 0 if X = 0, is one-to-one, actually bijective on R, but it is not strictly increasing, nor strictly decreasing or R. We shall return to this issue in Sect. 4.3. A useful remark is the following. The sum of functions that are similarly mono- tone (i.e., all increasing or all decreasing) is still a monotone function of the same kind, and turns out to be strictly monotone if one at least of the summands is. The map /(x) = x^ + x, for instance, is strictly increasing on R, being the sum of two functions with the same property. According to Proposition 2.8 / is then invertible; note however that the relation /(x) = y cannot be made explicit in the form X = f~^{y)' 2.5 Composition of functions Let X, y, Z be sets. Suppose / is a function from X to Y, and g a function from Y to Z. We can manifacture a new function h from X to Z by setting hix)^g{f{x)). (2.9) The function h is called composition of / and g, sometimes composite map, and is indicated by the symbol h — g o f (read ^g composed (with) / ' ). 44 2 Functions Example 2.9 Consider the two real maps y — f[x) = x — Z and z — g{y) — y'^ -\-1 oi one real variable. The composition of / and g reads z = h{x) —go f{x) = (x — 3)^ + 1. D Bearing in mind definition (2.9), the domain of the composition g o f is deter- mined as follows: in order for x to belong to the domain of ^ o / , f[x) must be defined, so x must be in the domain of / ; moreover, f(x) has to be a element of the domain of g. Thus X G dom g o f x E dom/ and f{x) E dom^. The domain of ^ o / is then a subset of the domain of / (see Fig. 2.10). Examples 2.10 X+2 ) The domain of f{x) = is R \ {1}, while g{y) — -yjy is defined on the x+2 interval [0, +CXD). The domain oi g o f[x) — consists of the x ^1 such \x-l\ x+2 that > 0; hence, dom^ o / = [—2, +(X)) \ {1}. ii) Sometimes the composition g o f has an empty domain. This happens for instance for f{x) = (notice f{x) < 1) and g{y) — ^Jy — h (whose domain 1 + x^ is [5,+00)). ^"f9{v") Figure 2.10. Representation of a composite function via Venn diagrams.  2.5 Composition of functions 45 The operation of composition is not commutative: if g o f and fog are b o t h defined (for instance, when X = Y = Z)^ the two composites do not coincide in 1 1 X general. Take for example f{x) = — and g{x) = -—^—^^ 1 for which go f{x) 1 but / o g(x) = 1 + X. If / and g are b o t h one-to-one (or b o t h onto, or b o t h bijective), it is not difficult to verify t h a t gof has the same property. In the first case in particular, the formula {g°f)-'^r'og-' holds. Moreover, if / and g are real monotone functions of real variable, gof too will be monotone, or better: g o f is increasing if b o t h / and g are either increasing or decreasing, and decreasing otherwise. Let us prove only one of these properties. Let for example / increase and g decrease; if xi < X2 are elements in d o m ^ o / , the monotone behaviour of / implies f{xi) < /(X2); now the monotonicity of g yields g{f{xi)) > g{f[x2)), so g o f is decreasing. We observe finally t h a t if / is a one-to-one function (and as such it admits inverse / ~ ^ ) , then / " ' o fix) = r\f{x)) =x, Vx e dom / , forHy) = firHy)) = y, Vyeim/. Calling i d e n t i t y m a p on a set X the function i d x : X ^ X such t h a t i d x {x) = x for all X G X , we have f~^ o f = iddom/ and / o f~^ = i d i m /. 2.5.1 T r a n s l a t i o n s , r e s c a l i n g s , r e f l e c t i o n s Let / be a real m a p of one real variable (for instance, the function of Fig. 2.11). Fix a real number c 7^ 0, and denote by tc : M -^ R the function tc{x) = x -\- c. Composing / with tc results in a t r a n s l a t i o n of the graph of / : precisely, the y = f{x) Figure 2.11. Graph of a function f{x) 46 2 Functions graph of the function fotc{x) = f{x-\-c) is shifted horizontally with respect to the graph of / : towards the left if c > 0, to the right if c < 0. Similarly, the graph of tc o / ( x ) = / ( x ) + c is translated vertically with respect to the graph of / , towards the top for c > 0, towards the bottom if c < 0. Fig. 2.12 provides examples of these situations. Fix a real number c > 0 and denote by Sc : M —^ R the map Sc{x) = ex. The composition of / with Sc has the effect of rescaling the graph of /. Precisely, if c > 1 the graph of the function / o sdx) = f{cx) is 'compressed' horizontally towards the y-axis, with respect to the graph of / ; if 0 < c < 1 instead, the graph 'stretches' away from the ^/-axis. The analogue effect, though in the vertical direction, is seen for the function Sc o f[x) — cf{x): here c > 1 'spreads out' the graph away from the x-axis, while 0 < c < 1 'squeezes' it towards the axis, see Fig. 2.13. Notice also that the graph of f{—x) is obtained by reflecting the graph of f{x) along the ^/-axis, like in front of a mirror. The graph of /(|x|) instead coincides with that of / for X > 0, and for x < 0 it is the mirror image of the latter with respect to the vertical axis. At last, the graph of |/(x)| is the same as the graph of / when / ( x ) > 0, and is given by reflecting the latter where /(x) < 0, see Fig. 2.14. , y = f{x + c),c>0 /(x + c), c < 0 /(x) + c, c > 0 y = f{x) + c, c < 0 Figure 2.12. Graphs of the functions /(x + c) (c > 0: top left, c < 0: top right), and f{x) + c (c < 0: bottom left, c > 0: bottom right), where f(x) is the map of Fig, 2,11  2.6 Elementary functions and properties 47 y = /(ex), c < 1 y = c/(x), c < 1 Figure 2.13. Graph of f{cx) with c > 1 (top left), 0 < c < 1 (top right), and of cf{x) with c > 1 (bottom left), 0 < c < 1 (bottom right) 2.6 Elementary functions and properties We start with a few useful definitions. Definition 2.11 Let f : d o m / C M -^ R 6e a map with a symmetric domain with respect to the origin, hence such that x G dom / forces —x G dom / as well. The function f is said even if f{—x) = f{x) for all x G d o m / , odd if f{—x) = —f{x) for all X G d o m /. The graph of an even function is symmetric with respect to the ^-axis, and that of an odd map symmetric with respect to the origin. If / is odd and defined in the origin, necessarily it must vanish at the origin, for /(O) = —/(O). Definition 2.12 A function f : dom f CR -^R is said periodic of period p (with p > 0 real) if dom / is invariant under translations by ±p (i.e., if X ± p £ dom/ for all x G domf) and if f{x 4- p) = f{x) holds for any X G dom/. 48 2 Functions y= f{-x) Figure 2.14. Clockwise: graph of the functions /(—x), /(|a:|), |/(|x|)|, | / ( x ) | One easily sees t h a t an / periodic of period p is also periodic of any multiple mp (m G N \ {0}) oi p. If the smallest period exists, it goes under the n a m e of m i n i m u m p e r i o d of the function. A constant m a p is clearly periodic of any period p> ^ and thus has no minimum period. Let us review now the main elementary functions. 2.6.1 P o v ^ e r s These are functions of the form y — x^. T h e case a = 0 is trivial, giving rise to t h e constant function y = x^ — \. Suppose then a > 0. For a = n G N \ {0}, we find the monomial functions y = x'^ defined on R, already considered in Example 2.7 ii) and iii). W h e n n is odd, the maps are odd, strictly increasing on M and with range R (recall P r o p e r t y 1.8). W h e n n is even, the functions are even, strictly decreasing on (—00,0] and strictly increasing on [0, +CXD); their range is the interval [0, +CXD). Consider now the case a > 0 rational. If a = ^ where m G N \ {0}, we define a function, called m t h root of x and denoted y — x^l"^ — A/X, inverting y — x^. It has domain R if m is odd, [0, +oo) if m is even. T h e m t h root is strictly increasing and ranges over R or [0, + o c ) , according to whether m is even or odd respectively. In general, for a — ^ G Q, n^m G N \ { 0 } with no common divisors, t h e function y — x^l'^ is defined as 2/ = {x^Yl'^ = \ / ^ - As such, it has domain R  2.6 Elementary functions and properties 49 Figure 2.15. Graphs of the functions y = x^^^ (left), y = x^^^ (middle) and y = x^^^ (right) if m is odd, [0, +oo) if m is even. It is strictly increasing on [0, +oo) for any n, m, while if m is odd it strictly increases or decreases on (—(X), 0] according to the parity of n. Let us consider some examples (Fig. 2.15). T h e m a p y = x^/^, defined on R, is strictly increasing and has range M. T h e m a p y = x^l'^ is defined on M, strictly decreases on (—oo, 0] and strictly increases on [0, + o o ) , which is also its range. To conclude, y — x^l^ is defined only on [0,+00), where it is strictly increasing and has [0, +00) as range. Let us introduce now the generic function y — x^ with irrational a > 0. To this end, note t h a t if a is a non-negative real number we can define the power a" with a G R + \ Q , starting from powers with rational exponent and exploiting the density of rationals inside M. If a > 1, we can in fact define oP" — s u p j a ' ^ / ^ | ^ < ce}, while for 0 < a < 1 we set a^ = inf{a"^/"^ | ^ < OL\. Thus the m a p y — x^ with a G M+ \ Q is defined on [0, + 0 0 ) , and one proves it is there strictly increasing and its range is [0, + 0 0 ). Summarising, we have defined y — x^ for every value a > 0. They are all defined a least on [0, + 0 0 ) , interval on which they are strictly increasing; moreover, they satisfy y(0) = 0, y(l) = 1. It will t u r n out useful to remark t h a t if a < /3, 0 < a:^ < x"" < 1, for 0 < X < 1, 1 < x^ < x^, for x > 1 (2.10) (see Fig. 2.16). i/Vs 0 I 1 Figure 2.16. Graphs of y = x^, x > 0 for some a > 0 50 2 Functions Figure 2.17. Graphs of y = x"^ for a two values a < 0 At last, consider the case of a < 0. Set y = x^ = by definition. Its x~^ domain coincides with the domain of ^ = x~^ minus the origin. All maps are strictly decreasing on (0, H-oo), while on (—cx), 0) the behaviour is as follows: writing a = —^ with m odd, the map is strictly increasing if n is even, strictly decreasing if n is odd, as shown in Fig. 2.17. In conclusion, we observe that for every a 7^ 0, the inverse function oi y = x^^ where defined, is y = x^l^. 2.6.2 Polynomial and rational functions A polynomial function, or simply, a polynomial, is a map of the form P{x) — anX^ ^ h a i x -h ao with a^ 7^ 0; ^ is the degree of the polynomial. Such a map is defined over all R; it is even (resp. odd) if and only if all coefficients indexed by even (odd) subscripts vanish (recall that 0 is an even number). P{x) A rational function is of the kind R{x) — , where P and Q are poly- Q[x) nomials. If these have no common factor, the domain of the rational function will be M without the zeroes of the denominator. 2.6.3 Exponential and logarithmic functions Let a be a positive real number. According to what we have discussed previously, the exponential function y — a^ \s defined for any real number x\ it satisfies ^(0) = aO = 1. If a > 1, the exponential is strictly increasing; if a = 1, this is the constant map 1, while if a < 1, the function is strictly decreasing. When a 7^ 1, the range is (0, +CXD) (Fig. 2.18). Recalling a few properties of powers is useful at this point: for any x, 2/ ^ M ^x+rw^*^ir 0*"^=? ^\^^^^ (a-r^ii m  2.6 Elementary functions and properties 51 1 2 3 Figure 2.18. Graphs of the exponential functions y = 2^ (left) and y = {^)^ (right) When a 7^ 1, the exponential function is strictly monotone on R, hence invertible. The inverse y = log^ x is called logarithm, is defined on (0, +oo) and ranges over R; it satisfies y{l) = log^ 1 = 0. The logarithm is strictly increasing if a > 1, strictly decreasing if a < 1 (Fig. 2.19). The previous properties translate into the following: 1 iogcti^y) -= log^ ^* -f logo y. yx,y>Q, X iogr l o g ^ ^ - - loga y. Va;, y > 0, logcX^') ==?/loga X, Vx > 0, Vi/ € E , 1 Figure 2.19. Graphs of y = log2 x (left) and y = log^/2 x (right) 2.6.4 Trigonometric functions and inverses Denote here by X, Y the coordinates on the Cartesian plane R^, and consider the unit circle, i.e., the circle of unit radius centred at the origin O = (0,0), whose 52 2 Functions equation reads X'^ + F^ = 1. Starting from the point A = (1,0), intersection of the circle with the positive x-axis, we go around the circle. More precisely, given any real x we denote by P{x) the point on the circle reached by turning counter-clockwise along an arc of length x if x > 0, or clockwise by an arc of length —X if X < 0. The point P{x) determines an angle in the plane with vertex O and delimited by the outbound rays from O through the points A and P(x) respectively (Fig. 2.20). The number x represents the measure of the angle in radians. The one-radian angle is determined by an arc of length 1. This angle measures ^ = 57.2957795 degrees. Table 2.1 provides the correspondence between degrees and radians for important angles. Henceforth all angles shall be expressed in radians without further mention. degrees 0 30 45 60 90 120 135 150 180 270 360 TT TT TT TT 27r 37r 57r 37r radians 0 TT 27r 6 4 3 2 3 4 6 2 Table 2.1. Degrees versus radians Increasing or decreasing by 27r the length x has the effect of going around the circle once, counter-clockwise or clockwise respectively, and returning to the initial point P(x). In other words, there is a periodicity P(x±27r) = F(x), VXG: (2.11) Denote by cosx ('cosine of x') and sinx ('sine of x') the X- and F-coordinates, respectively, of the point P(x). Thus P(x) = (cosx, sinx). Hence the cosine func- tion y = cosx and the sine function y = sinx are defined on R and assume all Figure 2.20. The unit circle  2.6 Elementary functions and properties 53 -27r /' \ —TT '\7: ITT 27r / Figure 2.21. Graph of the map y = sinx values of the interval [—1,1]; by (2.11), they are periodic maps of minimum period 27r. They satisfy the crucial trigonometric relation cos^ X + sin^ X = 1, Vx € R. (2.12) It is rather evident from the geometric interpretation t h a t the sine function is odd, while the cosine function is even. Their graphs are represented in Figures 2.21 and 2.22. Important values of these maps are listed in the following table (where k is any integer): TT sina: = 0 for x = fcTr, COS X = 0 for X = ~ 4- fcTT, TV sin x = 1 for X — -^ + 2k7r, cos X = 1 for X = 2fc7r, s i n x = —1 for x = ——-f2fc7r, c o s x = —1 for X = TT -f 2fc7r. v^...^-; / 27r Figure 2.22. Graph of the map y = cosx 54 2 Functions Concerning monotonicity, one has strictly increasing on - - + 2fc7r, - + 2/c7r y = smx IS < ^ ^, 37r ^, strictly decreasing on - + 2/c7r, —- + 2k7r strictly decreasing on [2fc7r, TT + 2A:7r] y = cos X IS strictly increasing on [TT + 2A:7r, 27r + 2/c7r]. The addition and subtraction formulas are relevant s m ( a ± /3) == sinacc^/? ± cosasin/3 Gos(of ± ^ = cos Of cos /? T sin a sin /?. Suitable choices of the arguments allow to infer from these the duplication formulas siii2^ == 2sHiar 0 and x -\- 5 ^ 0 are necessary. The first is tantamount to (x + l)(x — 4) > 0, hence x G (—oo, — 1] U [4, +oo); the second to X 7^ —5. The domain of / is then dom / = (-00, - 5 ) U ( - 5 , -1] U [4, +oo). c) d o m / - ( - 0 0 , 0 ) U(l,-foo). d) In order to study the domain of this piecewise function, we treat the cases X > 0, X < 0 separately. For X > 0, we must impose 2x + l 7^ 0, i.e., x 7^ — ^. Since — | < 0, the function is well defined on x > 0. For X < 0, we must have x + 1 > 0, o r x > —1. For negative x then, the function is defined on [—1,0). All in all, d o m / = [—l,+oo). 2. Ranges: a) The map y = x"^ has range [0,-foo); therefore the range of y = x^ + 1 is [1, +00). Passing to reciprocals, the given function ranges over (0,1]. b) The map is obtained by translating the elementary function y = y/x (whose range is [0, +00)) to the left by —2 (yielding y = \Jx + 2) and then downwards by 1 (which gives y — \/x -h 2 — 1). The graph is visualised in Fig. 2.27, and clearly mvj— [—1, +00). Alternatively, one can observe that 0 < \Jx + 2 < +00 implies — 1 < \Jx + 2 — 1 < +00, whence i m / = [—l,+oo). -1 Figure 2.27. Graph oiy = y/x + 2 - 1  2.7 Exercises 59 c) i m / = ( 0 , + o o ) ; d) i m / = ( - 7 , + o c ). 3. Imposing c o s x — 1 > 0 tells t h a t cosx > 1. Such constraint is true only for X — 2k7r, k e Z, where the cosine equals 1; thus d o m / = {x G R : x = 2/c7r, /c G Z } and i m / = {0}. Fig. 2.28 provides the graph. —GTT -47r -27r 0 27r 47r Qn Figure 2.28. Graph of y = Vcos x — 1 4. /-i([0,+ [4, + o c ) , we can compute / - I. [4, +oc) ^ (-00,1] , f^' : [4, +oo) ^ [1, +oo) explicitly. In fact, from x^ — 2x + 5 — y = 0 we obtain X = 1 ± x/2/-4. With the ranges of f^^ and /2~^ in mind, swapping the variables x, y yields f^\x) = 1 - x / ^ ^ , /2-Hx) = 1 + V ^ ^. 8. Since if X < 0, f{x)= { A/4^=^ if0 1 and the graph of / is in Fig. 2.32. The range of / is [—1, +(X)). To determine /~^ we discuss the cases 0 < x < 1 and X > 1 separately. For 0 < x < 1, we have — 1 < y < 8 and y = 9x^ - 1 y+1 For X > 1, we have y > S and y = 3x^ + 4x + 1 -2 + y3^TT Figure 2.32. Graph of y = (1 + 3x)(2x - \x - 1|) 62 2 Functions Thus x+l if - 1 < X < n{x) = { -2 + A/3X + 1. if X > 8. 10. Composite functions: a) As ^ o f{x) = g{f{x)) = g{x'^ - 3) = log(l + x^ - 3) = log(x2 - 2), it follows dom^ O / = { X G R : X 2 - 2 > 0 } = (-CX), - \ / 2 ) U (^/2, +oo). We have / o g{x) = f{g{x)) = /(log(l + x)) = (log(l + x ) ) ' - 3, so d o m / o ^ = { X G M : 1 + X > 0 } = (-1,H-OC). 2-5x b) gof{x) and dom^o/ = (-l,f]; x+1 7V2-^ and d o m / o ^ = (—00,2]. V2^^+l 11- 9{x) = ^ 2 _ ^ ^ and /i(x) = ^ o / ( x ). 12. After drawing the parabolic graphs / ( x ) and ^(x) (Fig. 2.33), one sees that x^ - 3x + 2 if X < 2 , h{x) = , 2. x^ - 5x + 6 if X > 2 , 1 2 3 Figure 2.33. Graphs of the parabolas f{x) = x^ — 3x -\-2 and g{x) = x^ — 5x -\- 6  2.7 Exercises 63 1 — — 3 I I 3 Figure 2.34. Graphs of the maps h (left) and k (right) relative to Exercise 12 and the graph of h is that of Fig. 2.34, left. Proceeding as above, 2:2 - 3x + 2 if ^ < 1 ^ k{x) 0 if 1 < X < 3 , a:^ — 5x + 6 if x > 3 , and k has a graph as in Fig. 2.34, right. Limits and continuity I This chapter tackles the limit behaviour of a real sequence or a function of one real variable, and studies the continuity of such a function. 3.1 Neighbourhoods The process of defining limits and continuity leads to consider real numbers which are 'close' to a certain real number. In equivalent geometrical jargon, one considers points on the real line 'in the proximity' of a given point. Let us begin by making mathematical sense of the notion of neighbourhood of a point. Definition 3.1 Let xo e R he a point on the real line, and r > 0 a real number. We call neighbourhood of Xo of radius r the open and bounded interval Ir{xo) =-- {xo -- r, Xo + r) = { X G M : |x - xo\ < r}. Hence, the neighbourhood of 2 of radius 10~^, denoted /io-i(2), is the set of real numbers lying between 1.9 and 2.1, these excluded. By understanding the quantity |x — xo| as the Euclidean distance between the points XQ and x, we can then say that /r(xo) consists of the points on the real line whose distance from xo is less than r. If we interpret |x — xo| as the tolerance in the approximation of xo by X, then /r(xo) becomes the set of real numbers approximating xo with a better margin of precision than r. Xo - r Xo Xo -\-r F i g u r e 3. 1. Neighbourhood of xo of radius r 66 3 Limits and continuity I Varying r in the set of positive real numbers, while mantaining XQ in R fixed, we obtain a family of neighbourhoods of XQ. Each neighbourhood is a proper subset of any other in the family that has bigger radius, and in turn it contains all neighbourhoods of lesser radius. R e m a r k 3.2 The notion of neighbourhood of a point XQ G R is nothing but a particular case of the analogue for a point in the Cartesian product R^ (hence the plane if d = 2, space if c? = 3), presented in Definition 8.11. The upcoming definitions of limit and continuity, based on the idea of neigh- bourhood, can be stated directly for functions on R^, by considering functions of one real variable as subcases for d = 1. We prefer to follow a more gradual ap- proach, so we shall examine first the one-dimensional case. Sect. 8.5 will be devoted to explaining how all this generalises to several dimensions. It is also convenient to include the case where XQ is one of the points -hcxD or — oo. Definition 3.3 For any real a > Q, we call neighbourhood of +oc with end-point a the open, unbounded interval Ia{+oo) = (a,-fcx)). Similarly, a neighbourhood of —oo with end-point —a will be defined as /a(-oc) = ( - 0 0 , - a ). —cxD —a 0 a +CXD Figure 3.2. Neighbourhoods of —CXD (left) and +oo (right) The following convention will be useful in the sequel. We shall say that the property P{x) holds 'in a neighbourhood' of a point c (c being a real number XQ or H-oc, — oo) if there is a certain neighbourhood of c such that for each of its points X, P{x) holds. Colloquially, one also says 'P(x) holds around c\ especially when the neighbourhood needs not to be specified. For example, the map f{x) = 2a: — 1 is positive in a neighbourhood of XQ = 1; in fact, f{x) > 0 for any x G /i/2(l). 3.2 Limit of a sequence Consider a real sequence a:n\-^ a^ We are interested in studying the behaviour of the values a^ as n increases, and we do so by looking first at a couple of examples.  3.2 Limit of a sequence 67 Examples 3.4 i) Let dn — n. The first terms of this sequence are presented in Table 3.1. We n +1 see that the values approach 1 as n increases. More precisely, the real number 1 can be approximated as well as we like by a^ for n sufficiently large. This clause is to be understood in the following sense: however small we fix s > 0, from a certain point n^ onwards all values a^ approximate 1 with a margin smaller that The condition la^ - 1| < £, in fact, is tantamount to < e, i.e., n + 1 > - ; n+ 1 e thus defining n^ = r and taking any natural number n > ris^we have n + 1 > 1 -1 > - , hence |a^ — 1| < e. In other words, for every s > 0, there exists an ris such that n > Tie ^ la^ — 1| < S- Looking at the graph of the sequence (Fig. 3.3), one can say that for all n > Ue the points (n, a^) of the graph he between the horizontal lines y = \ — e and n an n Cin 0 0.00000000000000 1 2.0000000000000 1 0.50000000000000 2 2.2500000000000 2 0.66666666666667 3 2.3703703703704 3 0.75000000000000 4 2.4414062500000 4 0.80000000000000 5 2.4883200000000 5 0.83333333333333 6 2.5216263717421 6 0.85714285714286 7 2.5464996970407 7 0.87500000000000 8 2.5657845139503 8 0.88888888888889 9 2.5811747917132 9 0.90000000000000 10 2.5937424601000 10 0.90909090909090 100 2.7048138294215 100 0.99009900990099 1000 2.7169239322355 1000 0.99900099900100 10000 2.7181459268244 10000 0.99990000999900 100000 2.7182682371975 100000 0.99999000010000 1000000 2.7182804691564 1000000 0.99999900000100 10000000 2.7182816939804 10000000 0.99999990000001 100000000 2.7182817863958 100000000 0.99999999000000 Table 3.1. Values, estimated to the 14th digit, of the sequences an (left) and 1+ (right) 68 3 Limits and continuity I l +e 1 1-s Figure 3.3. Convergence of the sequence an = ^^^ / 1\ ii) The first values of the sequence a^ = 1 H— are shown in Table 3.1. One could imagine, even expect, that as n increases the values a^ get closer to a certain real number, whose decimal expansion starts as 2.718... This is actually the case, and we shall return to this important example later. D We introduce the notion of converging sequence. For simplicity we shall assume the sequence is defined on the set {n G N : n > TIQ} for a suitable no > 0. Definition 3.5 A sequence a : n 1—> a^ convergesto the limit £ e R\or converges to £ or has limit i), in symbols lim On = e, n-^oo if for any real number e > 0 there exists an integer n^ such that Vn > no, n > n^ =^ \an - £\ n^ can be written n G /-^^(+oo), while \an — £\ < £ becomes a^ G Ie{£)- Therefore, the definition of convergence to a limit is equivalent to: for any neighbourhood Ie{£) of ^, there exists a neighbourhood In^{~\-oc) of +CXD such that 'in>nQ, nG/n^(+oc) =^ an^Iei^)' E x a m p l e s 3.6 i) Referring to Example 3.4 i), we can say lim - ^ = 1. n^oo n + 1  3.2 Limit of a sequence 69 ii) Let us check that lim —-7Z = 0. Given e > 0, we must show 3n < e 2 + 5n2 I for all n greater than a suitable natural number n^. Observing that for n > 1 3n 3n 3n 3 2 + 5n2 2 + 5n^ < 5n^ 5n' we have 3n 5n 2 + 5n2 < e. But < £ n > 5e' 5n so we can set n^ = D Let us examine now a different be- haviour as n increases. Consider for n dn instance the sequence 0 0 a : n ^-^ an = n^' 1 1 2 4 Its first few values are written in Ta- 3 9 ble 3.2. Not only the values seem not 4 16 to converge to any finite limit £, they 5 25 are not even bounded from above: 6 36 however large we choose a real num- 7 49 ber A > 0, if n is big enough (meaning 8 64 larger than a suitable n^), cin will be 9 81 bigger than A. In fact, it is sufficient 10 100 to choose UA — [\/^] and note 100 10000 1000 1000000 n> UA =^ n> VA ^ n^ > A. 10000 100000000 100000 10000000000 One says that the sequence diverges to +(X) when that happens. Table 3.2. Values of an In general the notion of divergent sequence is defined as follows. 70 3 Limits and continuity I Definition 3.7 The sequence a : n v-^ an tends to +oo {or diverges t o -foo, or has limit +oo), written lim an = +00, n—>oo if for any real A > 0 there exists an UA such that Vn > no, n> UA =^ an> A. (3.1) Using neighbourhoods, one can also say that for any neighbourhood IA{-^OO) of +00, there is a neighbourhood InA{+oo) of +(X) satisfying Vn > no, neInA{+^) => an G/A(+oc). The definition of lim an = —oo is completely analogous, with the proviso that the implication of (3.1) is changed to Vn > no, n> UA => ^n < —A. E x a m p l e s 3.8 i) From what we have seen it is clear that lim n = +CX). n—^ oo ii) The sequence a^ = 0 + 1 + 2 +... + n = \ J k associates to n the sum of the k=0 natural numbers up to n. To determine the limit we show first of all that n(n + l) (3.2) k=0 a relation with several uses in Mathematics. For that, note that a^ can also be written as an = n + (n — 1) +... + 2 + 1 + 0 = y ^ ( ^ — k), hence k=0 2an = y ^ k + /_](n — k) = Y^ n = n Y"^ 1 = n{n + 1), k=0 k=0 k=0 k=0 jiffi -]- \\ Tiin + 1) n and the claim follows. Let us verify lim = +(X). Since > —-, n^cx) 2 2 2 we can proceed as in the example above, so for a given A > 0, we may choose UA = [V2A] D  3.2 Limit of a sequence 71 The previous examples show that some sequences are convergent, other di- vergent (to +00 or —CXD). But if neither of these cases occurs, one says that the sequence is indeterminate. Such are for instance the sequence a^ = (—1)^, which we have already met, or /, / ,.r>\ f 2n for n even, A sufficient condition to avoid an indeterminate behaviour is monotonic- ity. The definitions concerning monotone functions, given in Sect. 2.4, apply to sequences, as well, which are nothing but particular functions defined over the natural numbers. For them they become particularly simple: it will be enough to compare the values for all pairs of subscripts n, n + 1 belonging to the domain of the sequence. So, a sequence is monotone increasing if Vn>no, an UQ, then the sequence converges to the supremum £ of its image: lim an — i = sup {an n > no};