Analysis · Hierarchy

Continuity says that small changes in input cause small changes in output, and uniform continuity asks for this at the same scale everywhere. A continuous function may need a smaller $\delta$ at some points than at others. A uniformly continuous function has one $\delta$ that works everywhere.

uniformly continuous ⊂ continuous

Definitions

continuous

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
Non-examples
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).

uniformly continuous

High school

A function $f : \reals \to \reals$ is uniformly continuous if for every $\varepsilon > 0$, there exists $\delta > 0$ such that

\[\abs{f(x) - f(y)} < \varepsilon \quad \text{for all } \abs{x - y} < \delta.\]

The choice of $\delta$ depends only on $\varepsilon$, so the same $\delta$ works at every point.

Undergraduate (metric spaces)

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

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

The choice of $\delta$ depends only on $\varepsilon$, so the same $\delta$ works at every point.

Examples
Non-examples
  • $f(x) = x^2$ on $\reals$
    $f(x) = x^2$ is not uniformly continuous on $\reals$, because its graph gets steeper and steeper, so no single closeness of inputs works everywhere.
  • $x \mapsto e^x$ on $\reals$
    $x \mapsto e^x$ is not uniformly continuous, because its graph gets steeper without bound as $x$ increases.
  • $x \mapsto \tan(x)$ on $(-\pi/2, \pi/2)$
    $\tan$ is not uniformly continuous there, because its graph gets steeper without bound near the ends of the interval.
  • $x \mapsto \sin(1/x)$ on $(0, 1]$
    $x \mapsto \sin(1/x)$ on $(0, 1]$ is not uniformly continuous, because points that are arbitrarily close together have values that differ by $2$. For each natural number $n$, let \[x_n = \frac{1}{\frac{\pi}{2} + 2 \pi n} \quad \text{and} \quad y_n = \frac{1}{\frac{3\pi}{2} + 2 \pi n}.\] Then $\sin(1/x_n) = 1$ and $\sin(1/y_n) = -1$, so $\abs{\sin(1/x_n) - \sin(1/y_n)} = 2$ for every $n$, but $\abs{x_n - y_n} \to 0$ as $n \to \infty$. Therefore, for $\varepsilon = 1$, no $\delta > 0$ satisfies the definition of uniform continuity: some pair $x_n, y_n$ is closer than $\delta$, but the values of the pair differ by more than $\varepsilon$.
  • $x \mapsto \sin(1/x)$ on $(0, \infty)$
    $x \mapsto \sin(1/x)$ is not uniformly continuous, because it oscillates between $-1$ and $1$ faster and faster as $x \to 0$.
Properties
  • Every uniformly continuous function is continuous. On a compact domain, every continuous function is uniformly continuous (the Heine–Cantor theorem).
  • A uniformly continuous function maps Cauchy sequences to Cauchy sequences.

Examples

Example continuous uniformly continuous Why
$f(x) = x^2$ on $\reals$ ✓✗ The squaring function, as a function $f : \reals \to \reals$.
$x \mapsto e^x$ on $\reals$ ✓✗
$x \mapsto \tan(x)$ on $(-\pi/2, \pi/2)$ ✓✗
$x \mapsto \sin(1/x)$ on $(0, \infty)$ ✓✗
$x \mapsto \sin(x)$ on $\reals$ ✓✓
$x \mapsto \sqrt{x}$ on $[0, \infty)$ ✓✓
$x \mapsto x^2$ on $[0, 1]$ ✓✓

Notes

  • Every uniformly continuous function is continuous (as the name suggests), but some continuous functions are not uniformly continuous.
  • Uniform continuity is defined for functions between metric spaces (or, more generally, uniform spaces), whereas continuity is defined for functions between all topological spaces.
  • To distinguish continuity from uniform continuity, the former is sometimes called “ordinary continuity” or “pointwise continuity.”
  • Uniform continuity is a global property, whereas continuity is a local property.
  • On a compact domain, every continuous function is uniformly continuous (the Heine–Cantor theorem).

See also

All distinctions · All definitions