Uniform Continuity between Metric Spaces¶

Definition¶

Consider metric spaces $(X,d_{X})$ and $(Y,d_{Y})$ , and a function $f\colon X\to Y$ $Y$ . We say $f$ is uniformly continuous on $X$ if, for any $\varepsilon > 0$ $0$ , there exists a $\delta > 0$ such that for all $x_1,x_2\in X$ with $d_{X}(x_1,x_2)<\delta$ $δ$ , it holds that $d_{Y}(f(x_1),f(x_2))<\varepsilon$ $ε$ .

Interpretation¶

Uniform continuity asks for one $\delta$ that works at every point of $X$ at once: inputs within $\delta$ of each other, wherever they lie, have outputs within $\varepsilon$ of each other. This is the general form of uniform continuity, with possibly different metrics on different spaces.

Two spaces X and Y with an arrow f. Balls of the same radius delta around three points in X are mapped inside balls of radius epsilon around their images in Y.

For a given $\varepsilon>0$ $0$ , the three green balls in $X$ all have the same radius $\delta$ , and each is mapped inside the $\varepsilon$ -ball around its centre's image. The same $\delta$ serves at every other point of $X$ .

The difference from continuity lies in the order of the quantifiers. For continuity, $\delta$ is chosen after the point and may depend on it; for uniform continuity, $\delta$ depends on $\varepsilon$ alone. Every uniformly continuous function is therefore continuous, but not conversely. The function $f(x)=1/x$ on $(0,1)$ is continuous, yet for $\varepsilon=1$ no $\delta$ works: for $0<x<\min\{2\delta,1\}$ the points $x$ and $x/2$ are less than $\delta$ apart, while $|f(x)-f(x/2)|=1/x>1$ $1$ .

Compactness removes the difference: a continuous function on a compact set is uniformly continuous, as the Key Results state, and for $[a,b]$ this is the Heine-Cantor theorem. Uniform continuity is what is needed when one estimate must hold over a whole domain, for example in the proof that a continuous function on $[a,b]$ is Riemann integrable.

A common slip is to speak of uniform continuity at a point. Uniform continuity is a property of a function on a set, since it compares all pairs of points of that set; at a single point there is only continuity.

Key Results¶

Any continuous function defined on a compact set is uniformly continuous.

Examples¶

Every constant function is uniformly continuous regardless of the metrics.
Every function from a discrete space is uniformly continuous.

Prerequisites¶

Continuity between Metric Spaces

Sources¶

Math III: Analysis · Lecture 9

Symbols: delta, element of, epsilon

Last updated: 20 October 2026 03:02