Uniform continuity

Let $f:(X,d_X)\to(Y,d_Y)$ and $A\subseteq X$. The function $f$ is uniformly continuous on $A$ if

Compared to ordinary continuity on $A$, the key point is that $\delta$ depends only on $\varepsilon$, not on the location in $A$.

Useful properties:

• Uniformly continuous functions send Cauchy sequences to Cauchy sequences; this links naturally with completeness.
• If $f$ is continuous on a compact set $K$, then $f$ is uniformly continuous on $K$ (Heine–Cantor).
• Every Lipschitz function (i.e., $d_Y(f(x),f(y))\le L\,d_X(x,y)$) is uniformly continuous.

Examples in $\mathbb{R}$:

• $f(x)=x^2$ is not uniformly continuous on $\mathbb{R}$, but it is uniformly continuous on $[0,1]$.
• $f(x)=1/x$ is continuous on $(0,1)$ but not uniformly continuous there.