Continuous Functions

Continuity is the central concept of analysis. A function is continuous if small changes in input produce small changes in output — equivalently, if the limit of the function at a point equals the function's value. Continuous functions preserve convergence, map compact sets to compact sets, and satisfy the Intermediate and Extreme Value Theorems.

Epsilon-delta definition

Definition4.1Continuity at a point

Let $f : D \to \mathbb{R}$ where $D \subseteq \mathbb{R}$ , and let $c \in D$ . Then $f$ is continuous at $c$ if for every $\epsilon > 0$ , there exists $\delta > 0$ such that

$|x - c| < \delta \quad \text{and} \quad x \in D \implies |f(x) - f(c)| < \epsilon.$

RemarkIntuition

Continuity at $c$ means: given any tolerance $\epsilon$ for the output, we can find a tolerance $\delta$ for the input such that inputs within $\delta$ of $c$ produce outputs within $\epsilon$ of $f(c)$ .

Definition4.2Continuous function

$f$ is continuous on $D$ if $f$ is continuous at every point $c \in D$ .

ExamplePolynomials are continuous

Every polynomial $p(x) = a_n x^n + \cdots + a_1 x + a_0$ is continuous on $\mathbb{R}$ . For $p(x) = x^2$ , given $\epsilon > 0$ and $c \in \mathbb{R}$ , let $\delta = \min(1, \epsilon/(2|c| + 1))$ . Then for $|x - c| < \delta$ ,

$|x^2 - c^2| = |x - c| \cdot |x + c| < \delta \cdot (2|c| + 1) \leq \epsilon.$

ExampleDiscontinuous function

The function

$f(x) = \begin{cases} 1 & \text{if } x \geq 0, \\ 0 & \text{if } x < 0 \end{cases}$

is not continuous at $x = 0$ . For any $\delta > 0$ , taking $x = -\delta/2$ gives $|x - 0| < \delta$ but $|f(x) - f(0)| = |0 - 1| = 1$ , so no $\delta$ works for $\epsilon = 1/2$ .

Sequential characterization

Theorem4.1Sequential continuity

$f$ is continuous at $c$ if and only if for every sequence $(x_n)$ in $D$ with $x_n \to c$ , we have $f(x_n) \to f(c)$ .

ProofSketch

( $\Leftarrow$ ) Suppose $f$ is not continuous at $c$ . Then there exists $\epsilon > 0$ such that for every $\delta > 0$ , there exists $x$ with $|x - c| < \delta$ and $|f(x) - f(c)| \geq \epsilon$ . Taking $\delta = 1/n$ , we get a sequence $(x_n)$ with $|x_n - c| < 1/n$ (so $x_n \to c$ ) but $|f(x_n) - f(c)| \geq \epsilon$ for all $n$ . Thus $f(x_n) \not\to f(c)$ , contradiction.

■

ExampleUsing sequential continuity

To show $f(x) = 1/x$ is continuous on $(0, \infty)$ , let $x_n \to c$ with $c > 0$ . Then

$\left|\frac{1}{x_n} - \frac{1}{c}\right| = \frac{|x_n - c|}{|x_n c|} \to 0$

(since $x_n \to c > 0$ , so $x_n$ is bounded away from zero eventually). Thus $f(x_n) \to f(c)$ .

Algebra of continuous functions

Theorem4.2Algebra of continuous functions

If $f, g : D \to \mathbb{R}$ are continuous at $c$ , then:

$f + g$ is continuous at $c$ .
$f \cdot g$ is continuous at $c$ .
If $g(c) \neq 0$ , then $f/g$ is continuous at $c$ (on the domain where $g \neq 0$ ).
If $h : \mathbb{R} \to \mathbb{R}$ is continuous at $f(c)$ , then $h \circ f$ is continuous at $c$ .

ExampleRational functions

Every rational function $r(x) = p(x)/q(x)$ (where $p, q$ are polynomials) is continuous on its domain $\{x \mid q(x) \neq 0\}$ .

ExampleComposition of continuous functions

Since $\sin$ and $x^2$ are continuous, $\sin(x^2)$ is continuous. Similarly, $e^{\sin(x)}$ , $\ln(1 + x^2)$ , etc., are continuous wherever defined.

Topological characterization

Theorem4.3Topological continuity

$f : D \to \mathbb{R}$ is continuous if and only if for every open set $U \subseteq \mathbb{R}$ , the preimage $f^{-1}(U) = \{x \in D \mid f(x) \in U\}$ is open in $D$ (with the subspace topology).

RemarkImportance

This characterization generalizes to arbitrary topological spaces and is the definition of continuity in topology. It says continuous functions are precisely those that preserve openness "backwards."

ExamplePreimages under continuous functions

Let $f(x) = x^2$ . Then $f^{-1}((1, 4)) = \{x \mid 1 < x^2 < 4\} = (-2, -1) \cup (1, 2)$ , which is open. Indeed, preimages of open intervals under $f$ are unions of open intervals.

Summary

Continuity is the fundamental property for functions in analysis:

Epsilon-delta definition: small changes in input produce small changes in output.
Sequential definition: preserves limits of sequences.
Algebraic operations and composition preserve continuity.
Topological definition: preimages of open sets are open.

See Intermediate Value Theorem and Extreme Value Theorem for major applications.