Continuous function

Definition · applies to functions

Elementary

Think of a function $y = f(x)$. The graph of $f(x)$ is a plot that shows the value of $y$ for each values of $x$ by plotting the value of $x$ on a horizontal axis and the value $y = f(x)$ on the vertical axis. The function is continuous if you can draw its graph without lifting your pencil.

Caveat. This only works when the domain is one unbroken interval. The function $x \mapsto 1/x$ is continuous, but its graph comes in two pieces because $0$ is not in its domain.

High school

Consider a function $f$. The function $f$ is continuous if

\[\lim_{x \to a} f(x) = f(a)\]

for every $a$ in the domain of $f$.

Undergraduate (metric spaces)

Consider a function $f : X \to Y$ between metric spaces. The function $f$ is continuous if for every $x \in X$ and every $\varepsilon > 0$, there exists $\delta > 0$ such that

\[d(f(x), f(y)) < \varepsilon \quad \text{for every } y \in X \text{ with } d(x, y) < \delta.\]

The choice of $\delta$ may depend on both $\varepsilon$ and $x$.

Graduate (topological spaces)

Consider a function $f : X \to Y$ between topological spaces. The function $f$ is continuous if the preimage $f^{-1}(V)$ is open in $X$ for every open set $V$ in $Y$.

Examples

Examples

• $f(x) = x^2$ on $\reals$ High school
  $x \mapsto x^2$ is continuous.
• $x \mapsto x^2$ on $[0, 1]$ High school
  $x \mapsto x^2$ on $[0, 1]$ is continuous.
• $f(x) = x^2 \sin(1/x)$ for $x \neq 0$, $f(0) = 0$, on $[-1, 1]$ Undergraduate
• The Cantor function on $[0, 1]$ Undergraduate
  The Cantor function is continuous, and uniformly continuous because $[0, 1]$ is compact.
• $x \mapsto \sin(1/x)$ on $(0, 1]$ Undergraduate
• $x \mapsto \sqrt{x}$ on $[0, 1]$ Undergraduate
  $x \mapsto \sqrt{x}$ on $[0, 1]$ is absolutely continuous because it is the integral of its derivative, so it is also uniformly continuous and continuous.
• $x \mapsto x^3$, from $\reals$ to $\reals$ Undergraduate
  $x \mapsto x^3$ is a homeomorphism of $\reals$: it is continuous, with the continuous inverse $x \mapsto \sqrt[3]{x}$. So it maps open sets to open sets and closed sets to closed sets.

Non-examples

Things that may be mistaken for examples.

• $f(x) = \frac{\sin x}{x}$, at $x = 0$ High school
  The function is not continuous at $x = 0$ because $f(0)$ is undefined, so $x = 0$ is not in the domain of $f$.
• The step function: $0$ for $x < 0$ and $1$ for $x \geq 0$ Undergraduate

Properties

• Sums, products, and compositions of continuous functions are continuous.
• Continuous images of compact sets are compact, and continuous images of connected sets are connected.
• A continuous function on $[a, b]$ takes every value between $f(a)$ and $f(b)$ (the intermediate value theorem).

Distinctions involving this term

• Topology vs. σ-algebra: Topologies and σ-algebras are both collections of subsets that are closed under some operations. A topology is closed under arbitrary unions and finite intersections, and describes nearness. A σ-algebra is closed under complements and countable unions, and describes what can be measured.
• Types of continuity: There are many ways to describe the continuity of a function. On a compact interval, each is a special case of the next, from continuously differentiable to continuous.

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