Uniform Continuity
Definition 1
- $(X, d_{X})$,
- metric sp.
- $(Y, d_{Y})$,
- metric sp.
- $f: X \rightarrow Y$
$f$ is said to be uniformly continuous if
\[\forall \epsilon > 0, \ \exists \delta > 0 \text{ s.t. } \forall x_{1}, x_{2} \in X, \ d_{X}(x_{1}, x_{2}) < \delta \Rightarrow d_{Y}(f(x_{1}), f(x_{2})) < \epsilon .\]■