INF 07 Continuous Functions PDF

# 7. Continuous functions ## Definitions * **Continuity**: * A function *f* is continuous at a point *a ∈ Df* * A function *f* is continuous (meaning continuous at every point in its domain) * A function *f* is continuous in set *A ⊆ Df* (meaning *f* is continuous at every point in *A*...

# 7. Continuous functions ## Definitions * **Continuity**: * A function *f* is continuous at a point *a ∈ Df* * A function *f* is continuous (meaning continuous at every point in its domain) * A function *f* is continuous in set *A ⊆ Df* (meaning *f* is continuous at every point in *A*) * Note: some authors assume that *f* is continuous in *A ⊂ Df* only if *f* is continuous at every point of set *A*. These definitions are not equivalent (they imply nuances) * **Continuity at a point** * **Heine's definition**: Let *f : D → Y*, *D ⊆ X*, where *(X, dx)*, *(Y, dy)* are metric spaces, and *a ∈ D*. We say that *f* is continuous at point *a* if and only if: * (∀{x_n}) [(x_n ∈ D, lim_{n→+∞} x_n=a) -> lim_{n→+∞} f(x_n) = f(a)] * Alternatively (equivalent): (∀ε › 0)(∃δ › 0)(∀x ∈ D) [dx(x, a) < δ -> dy(f(x), f(a)) < ε]. * **Cauchy's definition**: Let *f : D → Y*, *D ⊆ X*, where *(X, dx)*, *(Y, dy)* are metric spaces, and *a ∈ D*. We say that *f* is continuous at point *a* if and only if: * If "a ∈ D and a is an accumulation point of D", then "lim_{x→a} = f(a)" * If "a is isolated point of D", then we cannot calculate the limit of *f* at *a*. * (∀{x_n}) [(x_n ∈ D, lim_{n→+∞} x_n=a) -> lim_{n→+∞} f(x_n) = f(a)] * Alternatively (equivalent): (∀ ε > 0)(∃ δ > 0)(∀x ∈ D) [dx(x, a) < δ -> dy(f(x), f(a)) < ε]. * **Basic difference between limit and continuity**: * In the definition of the limit, *a* must be an accumulation point of *D*, but it doesn't need to be an argument of the function. * In the definition of continuity, *a* must be an argument of the function, but it doesn't need to be an accumulation point of *D*. * **Discontinuity**: if *a ∈ D* and *f* is not continuous at *a*, then we say that *f* is discontinuous at *a*. * **Continuity and domain**: continuity is defined only at the points in the domain. If *a ∉ D*, then *f* is neither continuous nor discontinuous at *a*. * **Continuity and accumulation point**: if *a ∈ D* and *a* is an accumulation point of *D*, then: * *"f is continuous at a"<-> *"lim_{x→a} f(x) = f(a)"*. * *"f is discontinuous at a"<-> *"lim_{x→a} f(x) ≠ f(a)"*. ## Types of discontinuity for real-valued function of one real variable * **Removable discontinuity (gap)**: *a* is a point of discontinuity, and the limit of *f* at *a* exists. We can easily remove this type of discontinuity (the resulting function will be continuous at *a*). * **Jump discontinuity**: both one-sided limits of the function exist at *a*, but they are different. * **Infinite jump**: at least one of the one-sided limits of *f* at *a* is not finite. * **Oscillatory discontinuity**: in every neighborhood of point *a*, the function takes all the values from a certain interval infinitely many times. * Example: *f(x) = sin(1/x)* has this kind of discontinuity at *x = 0*. * **Other discontinuity**: these discontinuities are not included in the previous classifications. * **Dirichlet function example**: Let *a ∈ ℝ*. We can always find two sequences {x_n}, {x_n'} converging to *a*, such that x_n ∈ ℚ and x_n' ∈ ℝ \ ℚ. In this case: * lim_{n→+∞} f(x_n)= lim_{n→+∞} 1 = 1 * lim_{n→+∞} f(x'_n)= lim_{n→+∞} 0 = 0 * If we take a sequence with infinitely many rational and irrational elements, the corresponding sequence of values of the Dirichlet function will be divergent (its limit doesn't exist, even though the sequence is bounded). Therefore, every point is a point of discontinuity of the second kind, belonging to the category "other". ## Theorems * **Theorem 1**: If the functions *f* and *g* are continuous at a point *a*, then the functions: * *k * f* (k ∈ℝ), * *f + g*, * *f * g*, * * f/g* (if *g ≠ 0*) are continuous at *a*. * **Theorem 2**: If *f* is continuous at *a* and *g* is continuous at *b = f(a)*, then *g o f* is continuous at *a*. * Proof: For example, if *a* and *b* are accumulation points of their domains: * lim_{x→a} f(x) = f(a) * lim_{y→b} g(y) = g(b) * Let *h(x) = g(f(x))*. Then: * lim_{x→a} h(x) = lim_{x→a} g(f(x)) = lim_{y→f(a)} g(y) = g(b) = g(f(a)) = h(a) * **Theorem 3**: The inverse of a continuous and strictly monotonic function is continuous in the interval where it is defined. * Note: this condition is sufficient, but not necessary. * Example: let *f: y = tan x, x ∈ (0, π/2) ∪ (π/2, π)* * Example: let *g: y = tan x, x ∈ (0, π/4) ∪ (π/4, π/2)* * *f* is not monotonic, *f⁻¹* is continuous in its domain. * *g* is not monotonic, *g⁻¹* is discontinuous at *x = 0* (has a jump discontinuity) * **Theorem 4**: Every elementary function is continuous. * Proof: * The functions *U*, *id*, *exp*, and *sin* are defined on ℝ. Every point of their domain is an accumulation point. We need to prove that lim_{x→a} f(x) = f(a) for every *a ∈ ℝ*: * lim_{x→a} U(x) = lim_{x→a} 1 = 1 = U(a) * lim_{x→a} id(x) = lim_{x→a} x = a = id(a) * lim_{x→a} exp(x) = lim_{x→a} e^x = e^a = exp(a) for every *a ∈ ℝ*. * We can prove that *exp* is continous at 0, *i.e.* * (∀ ε > 0)(∃ δ > 0)(∀ x ∈ ℝ) [|x| < δ -> |e^x - e^0| < ε] * (∀ ε > 0)(∃ δ > 0)(∀ x ∈ ℝ) [|x| < δ -> |e^x - 1| < ε] * (∀ ε > 0)(∃ δ > 0)(∀ x ∈ ℝ) [-δ < x < δ -> 1 - ε < e^x < 1 + ε] * We know that lim_{n→+∞} e^(1/n) = 1. Therefore: * (∀ ε > 0)(∃ N ∈ ℕ)(∀n > N) [1 - ε < e^(1/n) < 1 + ε] * (∀ ε > 0)(∃ N ∈ ℕ)(∀n > N) [1 - ε < e^(-1/n) < 1 + ε] (because the exponential function is increasing) * Let N = max{N_1, N_2}. If -δ < x < δ, then: * 1 - ε < e^(-δ) < e^x < e^(δ) < 1 + ε * 1 - ε < e^(x) < 1 + ε * Therefore, (*) holds because for every ε > 0, we can find N and choose δ = 1/N. * lim_{x→a} sin(x) = lim_{x→a} sin(x) = lim_{t→0} sin(a+t) = lim_{t→0} (sin(a)cos(t) + cos(a)sin(t)) = sin(a) + cos(a) * 0 = sin(a) = sin(a) * We proved that *sin* is continuous at *a* if and only if: * (∀ ε > 0)(∃ δ > 0)(∀ x ∈ ℝ) [|x-a| < δ -> |sin(x) - sin(a)| < ε] * |sin(x) - sin(a)| = 2 cos((x + a)/2) sin((x - a)/2) < 2 sin((x - a)/2) < 2 * ((x -a)/2) = x - a * Since |sin(x) - sin(a)| < |x - a| , it is enough to choose δ = ε. * Other elementary functions are continuous either because they are a composition of elementary functions, or because they are obtained from continuous functions by using operations that preserve continuity (sum, difference, product, quotient, composite function). * **Theorem 5 (sign preservation)**: If * f : (a, b) → ℝ* is continuous and *f(c) > 0* for some *c ∈ (a, b)*, then *f* takes only positive values in a certain neighborhood of point *c*. * **Theorem 6 (Darboux property)**: If *f : (a, b) → ℝ* is continuous, then for any *d* between *f(a)* and *f(b)*, there exists *c ∈ (a,b)* such that *f(c) = d*. * **Theorem 7 (intermediate value theorem)**: If *f : (a, b) → ℝ* is continuous and *f(a) *f(b) < 0*, then there exists *c ∈ (a, b)* such that *f(c) = 0*. * **Theorem 8 (Weierstrass)**: If *f : (a, b) → ℝ* is continuous, then *f* reaches its least upper bound M and greatest lower bound m. * **Corollary**: If *f* is continuous on *(a, b)*, then: * *Rf = (m, M)* (range of f) * *f* is bounded * **Definition 3**: The oscillation of the function f is the difference between its largest and smallest values: ω = M - m. * **Definition 4**: A partition of * (a, b)* is a set: * *P_n= {(x_0, x_1), (x_1, x_2), (x_2, x_3),..., (x_{n-1}, x_n)}*, where *a = x_0 < x_1 < x_2 < ... < x_n = b*. * **Theorem 9**: If *f* is continuous on *(a, b)*, then there exists a partition of *(a, b)* such that the oscillation in each subinterval is less than an arbitrary positive number: * (∀ ε > 0)(∃ n ∈ ℕ, P_n, 1 ≤ i ≤ n)[w_i = M_i - m_i < ε]