Theorems on Limits & Continuity PDF

# LESSON 20 ## LIMITS OF FUNCTIONS & CONTINUITY ### (THEOREMS ON CONTINUITY - THE n VARIABLE CASE) # LIMITS OF FUNCTIONS AND CONTINUITY ## (THEOREMS ON CONTINUITY - THE n VARIABLE CASE) ### THEOREM 10 WEIERSTRASS'S THEOREM Consider a function f: A=IR->IR. yf (i) A is a COMPACT SET IN IR → CLOSED...

# LESSON 20 ## LIMITS OF FUNCTIONS & CONTINUITY ### (THEOREMS ON CONTINUITY - THE n VARIABLE CASE) # LIMITS OF FUNCTIONS AND CONTINUITY ## (THEOREMS ON CONTINUITY - THE n VARIABLE CASE) ### THEOREM 10 WEIERSTRASS'S THEOREM Consider a function f: A=IR->IR. yf (i) A is a COMPACT SET IN IR → CLOSED AND BOUNDED (Li) f is CONTINUOUS in A → NOT NECESSARILY AN INTERVAL! Then, THERE EXISTS: max f(x)=M min f(x)=m in A Graphically, we may have many cases: - M - m - a - b - f - M - m - a - b - f - M - m - a - b - f - M=m - a - b - f WARNING! Domain A NEED NOT BE an interval [a, b]. For instance: - M - m - 1 - 4 - 6 - 8 - A=[1,4]U[6,8] - COMPACT, NOT AN INTERVAL! ### REMARK! Weierstress theorem is an important Result, with ADVANTAGES and DRAWBACKS #### ADVANTAGES: it is a theorem that guarantees, in some situations, the existence of (GLOBAL) MAXIMUM and MINIMUM. #### DRAWBACKS: the theorem just guarantees the existence of maximum and minimum, it DOES NOT Say where they are (they could be interior or boundary points and it DOES NOT give a procedure to find them! #### HOMEWORK: Show that theorem 10 does not hold if 1) A is bounded but not closed, or 2) A is closed but not bounded or 3) f is not continuous in A To do this, find COUNTEREXAMPLES. # THEOREM 11 BOLZANO'S THEOREM OR ZERO-VALUE THEOREM Consider a function f: A ÇIR->IR If (i) A is a COMPACT INTERVAL [a,b] (ii) f is CONTINUOUS in A (iii) f(a). f(b) ≤0 [f(a)≤0, f(b) ≤0 or f(a)>0, f(b)<0 Then: 1) ∃ c ∈ [a,b]: f(c)=0 2) if f is STRICTLY MONOTONIC => C is UNIQUE Graphically, we may have: - f(b) ↑ - f(a) - f(c) - a - c - b - f - f(b) ↑ - f(a) - f(c) - a - c - b - f - f(a)=f(b)=0 - a - b - f ### REMARK: Using Bolzano's theorem, we can prove an interesting result on the EXISTENCE OF MARKET EQUILIBRIA: #### THEOREM, Let [a, b] the set of prices; consider: - a DEMAND FUNCTION D(P): [a,b]→R - and a SUPPLY FUNCTION S(P): [a; b]→R Suppose that D and S are continuous and - D(a), S(a) (AT THE LEAST PRICE) - D(b)≤S(b) (AT THE MAXIMUM PRICE) Then, by the theorem 11, there exists a MARKET EQUILIBRIUM (p,q) with p∈ [a,b] This equilibrium is UNIQUE provided that: 1) D is STRICTLY decreasing (The higher the price, the lower the demand) 2) S is STRICTLY increasing (The higher the price, the higher the supply) - ↑ - S - ↑ - D - a - p - b # WEIERSTRSS'S THEOREM APPLICATION TO THE CONSUMER PROBLEM Weierstrass's theorem is often used in economics, particularly in consumer theory, to establish the existence of optimal consumption bundles. The theorem states that a continuous function defined on a compact set achieves its maximum and minimum values. ## Application in the Consumer Problem 1. **Consumer Preferences and Utility Function:** - A consumer's preferences can be represented by a utility function U (x₁, x₂), where x₁ and x₂ are quantities of two goods. - Assume that the utility function is continuous and quasi-concave. 2. **Budget Constraint:** - The consumer faces a budget constraint of the form p₁x₁ + p₂x₂ ≤ I, where p₁ and p₂ are the prices of the goods, and I is the income. 3. **Compact and Convex Set:** - The feasible consumption bundles (combinations of x₁ and x₂) can be represented as a set B defined by the budget constraint. - This set is compact (closed and bounded) and convex in R². 4. **Application of Weierstrass's Theorem:** - By Weierstrass's theorem, since the utility function U (x₁, x₂) is continuous on the compact set B, it must achieve a maximum value at some point in B. - This point corresponds to the optimal consumption bundle, which maximizes the consumers' utility given their budget constraint. 5. **Conclusion:** - The existence of an optimal consumption bundle implies that consumers will always find at least one combination of goods that maximizes their utility, given their budget constraints. This application illustrates how mathematical principles can underpin economic theories, ensuring that consumers have well-defined choices under given constraints. # THEOREM 12 DARBOUX'S THEOREM OR INTERMEDIATE - VALUE THEOREM Consider a function f: A=IR->IR (i) A is a COMPACT INTERVAL [a, b] (ii) f is CONTINUOUS in A Then, the following conditions hold: VLE [ min f(x), wuer f(x)], ∃ c ∈ [a,b]: f(c) = L Graphically: - mar f(a)=M - f - L - min f(a) = m - a - c - b VLE [m, M], the STRAIGHT LINE Y=L meets the graph of f. #### HOMEWORK: Show (by counterexample) that, if f is NOT CONTINUOUS, then Theorem 12 DOES NOT HOLD. ### REMARK: As a consequence of Weierstress's and Darbour's theorems, we have the following nice result:. * if f is continuous on a compact interval [a, b], then f ([a,b]) is the compact interval [m, M]. # THEOREM 13 STRICI MONOTONICITY IS EQUIVALENT TO INVERTIBILITY, FOR CONTINUOUS FUNCTIONS ON INTERVALS. Consider a function f: A ÇIR→R. (i) A is an INTERVAL → NOT REQUIRED TO BE CLOSED, OPEN, BOUNDED.. (ii) f is CONTINUOUS in A Then: f is INVERTIBLE <=> f is STRICTLY MONOTONIC in A iff in A ### REMARK: The implication ← always holds by theorem 2 (Function) If f is NOT CONTINUOUS, or A is NOT AN INTERVAL, then The implication ⇒ is FALSE! For instance, consider the function f: A ÇIR->IR, with A = [2,5] U [6,8] (NOT AN INTERVAL!) - f(x) - 1 - 2 - 5 - 6 - 8 f(x) is INJECTIVE and CONTINUOUS but NOT STRICTLY MONOTONIC! # LIMITS and CONTINUITY for f: ASIK" → R The motion of limits for f: AÇIR→IR, can be extruded to the case of functions f: AÇIR^n→IR. The DEFINITION (in terms of NEIGHBORHOODS) is basically The same! ## DEFINITION: LIMIT FOR f: IR^n→IR^n Consider a function f: AÇIR^n→IR^n and an ACCUMULATION POINT c∈IR^n for A (NO NEED THAT c ∈ A) Then: lim f (x) = e x→c means that: # neighborhood V(c) : ∃ a neighborhood V(c) such that x∈V(c) {c} => = f(x) ∈V(e) Thus the definition of limit, and all the theory are VERY SIMILAR to the case f: A ÇIR→R! But the exercises are more difficult = we will only consider functions of 2 VARIABLES, and 3 KINDS of EXERCISES: ## 1 NO INDETERMINATE FORMS For instance: - lim (x,y)→(0,0) x²+y² = +∞ - lim (x,y)→(1,0) x²+y² = 3 + x + y = 4 - lim (x,y)→(0,0) x²+y² = e^{x² + y²} = 0 ## 2) INDETERMINATE FORMS that can BE REDUCED TO 1-VARIABLE LIMITS. For instance: - a) lim (x,y)→(0,0) xy e^xy - 1 = 0 = lim t→0 t e^t - 1 = 1 =D (x,y)→(0,0) =Dt0 =Dt0 - Set t = xy = Dt0 =Dt0 =Dt0 = Fundamental Limit - b) lim (x,y)→(0,0) x²+ y² sin(x²+y²) = 0 = lim t→0 t sin(t) = 1 = Fundamental Limit ## 3) SOME EXERCISES, WHEN PREVIOUS PROCEDURES DO NOT WORK Consider - lim (x,y)→(0,0) y x = 0 = 0 Here we can claim that THIS LIMIT DOESN'T EXIST! To understand this, note that there are too many directions along which (x,y) (0,0)=0 Usually, if you change the direction also the result of the limit changes! - Suppose (x,y)→(0,0) ALONG y=x => lim (x,y)→(0,0) y x = lim (x,y)→(0,0) x x = (1 - All (x, y) such that y=x - Suppose (x,y)→(0,0) ALONG y=-x => lim (x,y)→(0,0) y x = lim (x,y)→(0,0) x x = (-1 - All (x,y) such that y=-x - We can conclude that - lim (x,y)→(0,0) y x DOES NOT EXIST! Another EXAMPLE of NON-EXISTENCE IS - lim (x,y)→(0,0) x + 3y x - 2y - If (x,y) → (0,0) ALONG y = 0) (all (x,y) such that y=0) - lim (x,y)→(0,0) x + 3y x - 2y = 1 - If (x,y)→(0,0) ALONG x=0) (all (x,y) such that x=0) - lim (x,y)→(0,0) x + 3y x - 2y = 3 = 2 → As for CONTINUITY, we have the following generalization of DEFINITIONS ## DEFINITION: CONTINUITY OF f: A ÇIR^n → IR AT X₀ Consider a function f: AÇIR^n→IR and a point X₀∈A, Then: 1) if X₀ is an ACCUMULATION POINT of A, we say that f is CONTINUOUS AT X₀ if - a) lim f(x)=l x→x₀ - b) l=f(x₀) 2) if x₀ is an ISOLATED POINT of A, we just say that f is CONTINUOUS AT X₀. ## DEFINITION: CONTINUITY OF f: A ÇIR^n→IR in A Consider a function f: AÇIR^n→IR, we say that f is CONTINUOUS in A if f is continuous at x₀ ∀x₀∈A ## INTUITIVELY: for functions f: AÇIR²→R, a continuous function is a function whose graph is a SURFACE with NO HOLES, JUMPS, Finally, many theorems on continuous functions f: A ÇIR→IR can be generalized, but sometimes this is NOT EASY! So, we just state two of them: # THEOREM 14 CONTINUITY OF ELEMENTARY FUNCTIONS Every ELEMENTARY FUNCTION f: AÇIR^n→IR IS CONTINUOUS on it's NATURAL Domain A. # THEOREM 15 WEIERSTRASS'S THEOREM FOR f: AÇIR^n→IR Consider a function f: AÇIR^n→IR (i) A is a COMPACT SET in R^n (ii) f is CONTINUOUS in A Then THERE EXIST: - max f(x)=M - min f(x)=m - in A ## REMARK: It is difficult to generalize Bolzano's and Darbour's Theorems, because they are only valid for COMPACT INTERVAL in R^n the generalization of the notion of INTERVAL IS The notion of CONVEX SET. At the present stage, we can only state the following nice result: * "If f: A ÇIR^n→IR is CONTINUOUS in a COMPACT and CONVEX set AÇIR^n, there f(A) is the COMPACT INTERVAL [m, M]." - A - COMPACT and CONVEX SET IN R^n - M - f - m - COMPACT and CONVEX SET IN IR