Uniform Continuity of Functions on Metric Spaces

# Uniform Continuity of Functions on Metric Spaces

Recall from the Continuity of Functions on Metric Spaces page that if $(S, d_S)$ and $(T, d_T)$ are metric spaces then a function $f : S \to T$ is said to be continuous at a point $p \in S$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that if $d_S(x, p) < \delta$ then $d_T(f(x), f(p)) < \epsilon$.

Furthermore, we said that if $A \subseteq S$ and if $f$ is continuous for all $a \in A$ then $f$ is said to be continuous on $A$.

We will now look at a similar notion known as uniform continuity of a function $f : S \to T$ which we define below.

 Definition: Let $(S, d_S)$ and $(T, d_T)$ be metric spaces, $A \subseteq S$, and $f : A \to T$. Then $f$ is said to be Uniformly Continuous on $A$ if for all $\epsilon > 0$ there exists a $\delta > 0$ such that for all $x, y \in A$ with $d_S(x, y) < \delta$ we have that $d_T(f(x), f(y)) < \epsilon$.

It is important to note that the domain $A$ of $f$ will often determine whether $f$ is uniformly continuous of not.