= Solution
{c}
A <function> $f:E\to\mathbb R$ is <continuous> on $E$ if it is <continuous> at every $p\in E$, meaning
$$
\boxed{\forall p\in E\ \forall\varepsilon>0\ \exists\delta>0\ \forall x\in E:\quad |x-p|<\delta\ \Longrightarrow\ |f(x)-f(p)|<\varepsilon.}
$$
The <function> has <uniform continuity> on $E$ when
$$
\boxed{\forall\varepsilon>0\ \exists\delta>0\ \forall x,y\in E:\quad |x-y|<\delta\ \Longrightarrow\ |f(x)-f(y)|<\varepsilon.}
$$
For <continuity>, $\delta$ may depend on both $p$ and $\varepsilon$. For <uniform continuity>, one $\delta$ must work for every pair of points in $E$.
Back to article page
