notes-archive/docs/math/real-analysis/5-continuity.md

Chapter 5 - Continuity

Section 5.1 - Continuous Functions

Definition. Let A \subseteq \mathbb{R}, and f: A \rightarrow \mathbb{R}. Then, if a \in A, f is continuous at $a$ if, given any \epsilon > 0, there exists some \delta > 0 such that for all x \in A,

|x - a| < \delta \Rightarrow |f(x) - f(a)| < \epsilon

Note that if a is an isolated point of A, that is, not a cluster point, then a is automatically continuous.

If a is a cluster point of A, then this definition collapses to the definition of \lim_{x \rightarrow a} f(x) = f(a).

Note that a function cannot be continuous at a point outside of its domain, even if the limit exists.

Definition. f is continuous on A if it is continuous at every point a \in A.

Theorem. f is continuous if and only if for every sequence (x_n) in A that converges to a, the sequence (f(x_n)) converges to f(a).

---

Definition. Let (S, d_S) and (T, d_T) be metric spaces. A function f: S \rightarrow T is continuous at a point a \in S if given any \epsilon > 0, there exists some \delta > 0 such that for all x \in S,

d_S(x, a) < \delta \Rightarrow d_T(f(x),  f(a)) < \epsilon

Theorem. A function f: S \rightarrow T is continuous at a point a \in A if and only if given some neighborhood V(f(a)) \in B, there exists some U(a) \in A such that f(U) \subseteq V.

Section 5.2 - Combinations of continuous Functions

Theorem. Let f, g: A \rightarrow \mathbb{R} be continuous at a \in A. Then,

• f + g and fg are continuous at a
• If g(x) \neq 0 for all x \in A, then \frac{f}{g} is continuous at a.

As a consequence, every polynomial, rational, and basic trigonometric function are continuous on its domains.

Theorem. Lett A, B \subseteq \mathbb{R}, such that f: A \rightarrow B and g: B \rightarrow \mathbb{R}. Then, if c is a cluster point of A such that \lim_{x \rightarrow c} f(x) = L \in B and g is continuous at L, then

\lim_{x \rightarrow c} g(f(x)) = g(L) = g(\lim_{x \rightarrow c} f(x))

Corollary. let A, B \subseteq \mathbb{R}, with f: A \rightarrow B and g: B \rightarrow \mathbb{R}. If f is continuous at a \in A and g is continuous at f(a) \in B, then g(f(x)) is continuous at a.

Section 5.3 - continuous functions on Intervals

Theorem. Let S, T be metric spaces with A \subseteq S and f: A \rightarrow T. If A is a compact subset of S, then f(A) is a compact subset of T.

Corollary. Let f: A \rightarrow \mathbb{R} be a continuous function, with A being a compact subset of metric space S. Then, f(A) is closed and bounded. Moreover, there exists a p, q \in A such that f(p) and f(q) are the supremum and infimum of f(A).

Corollary. Maximum-Minimum Theorem. If I = [a, b] is a closed and bounded interval and f: I \rightarrow \mathbb{R} is continuous on I, then f has an absolute minimum and maximum on I.

---

Theorem. Let S, T be metric spaces and A \subseteq S. Then, if f: A \rightarrow T is continuous on A, and A is a connected subset of S, then f(A) is a connected subset of T.

Corollary. Suppose that I is an interval. Let f: I \rightarrow \mathbb{R} be continuous on I. Then, f(I) is an interval.

Theorem. (Bolzano's) Intermediate Value Theorem. Suppose f: [a, b] \rightarrow \mathbb{R} is continuous on [a, b] with a \neq b. Then, given some k such that f(a) < k < f(b), there exists some c \in (a, b) such that k = f(c).

---

Definition. Let A \subseteq R. Then, a function f: A \rightarrow \mathbb{R} is uniformly continuous if given any \epsilon > 0, there exists some \delta > 0 depending only on \epsilon such that for any x, y \in A,

|x - y| < \delta \Rightarrow |f(x) - f(y)| < \epsilon

Note that if f is uniformly continuous, it must be continuous on A.

Theorem. Let I = [a, b] be a closed and bounded interval. If f: I \rightarrow \mathbb{R} is continuous on I, then f is uniformly continuous.

Remark. If S, T are metric spaces, K is a compact subset of S, and f: K \rightarrow T is continuous on K, then f is uniformly continuous.

Theorem. Suppose A \subseteq \mathbb{R} and f: A \rightarrow \mathbb{R} is uniformly continuous. Then, if (x_n) is a Cauchy sequence in A, (f(x_n)) is a Cauchy sequence in \mathbb{R}.

Remark. Suppose S, T are metric spaces and f: S \rightarrow T is uniformly continuous. Then, if (x_n) is a Cauchy sequence in S, (f(x_n)) is a Cauchy sequence in T.