Uniformly continuity

Definition

$f$ is uniformly continuous on an interval $I$ if $\forall ϵ>0$, $\exists \delta$, $\forall x,y$ such that $|x-y|<\delta ⟹|f(x)-f(y)|<ϵ$, where $\delta$ is independent of the choice of $x,y$.

Proposition

If $f$ is continuous on$[a,b]$, $f$ is uniformly continuous on $[a,b]$. (This proposition is actually a version of Heine–Cantor Theorem for real-valued function.)