Continuity

Overview

A function $f$ is said to be continuous at a point $p$ if $f$ is defined at $p$ and

lim_{x \to p} f (x) = f (p) .

A jump discontinuity is a discontinuity formed when the left- and right-hand limits both exist at a point but are not equal. An infinite discontinuity is a discontinuity in which the function takes arbitrarily large values at a point. A removable discontinuity is a discontinuity that could be removed by redefining the function at the given point.

Algebraic Limit Theorem

Since continuity is defined in terms of limits, it shares the same algebraic properties of limits.

Commutative Limits

If $L = lim_{x \to a} g (x)$ exists and is in the domain of a function $f$ continuous at $L$ , then

f (lim_{x \to a} g (x)) = lim_{x \to a} f (g (x))

Composition

Assume $v$ is continuous at $p$ and that $u$ is continuous at $v (p)$ . Then the composition function $f = u \circ v$ is continuous at $p$ .

Inversion

Let $f$ be a continuous strictly increasing function on $[a, b]$ . Let $c = f (a)$ and $d = f (b)$ . Then $g = f^{- 1}$ is continuous and strictly increasing on $[c, d]$ .

An analogous implication holds for continuous strictly decreasing functions.

Intermediate Value Theorem

Let $f$ be continuous on $[a, b]$ . Let $x, y \in [a, b]$ such that $x < y$ and $f (x) \neq f (y)$ . Then $f$ takes on every value between $f (x)$ and $f (y)$ in interval $(x, y)$ .

Bolzano's Theorem

Let $f$ be continuous on $[a, b]$ and assume $f (a)$ and $f (b)$ have opposite signs. Then there exists a $c \in (a, b)$ such that $f (c) = 0$ .

Extreme Value Theorem

Let $f$ be a continuous function on $[a, b]$ . Then there exist points $c, d \in [a, b]$ such that $f (c) = sup f$ and $f (d) = inf d$ .

Boundedness Theorem

Let $f$ be continuous on $[a, b]$ . Then $f$ is bounded on $[a, b]$ .

This is typically proven using the method of successive bisection. Recursively bisect the interval containing a discontinuity. The supremum of left endpoints (or infimum of right endpoints) is a discontinuous point.

Uniform Continuity

Let $f$ be continuous on $[a, b]$ . The span of $f$ in interval $[a, b]$ is the difference $sup f - inf f$ . Then for all $ϵ > 0$ , there exists a finite partition of $f$ such that the span of $f$ in every subinterval is less than $ϵ$ .