Continuity of a function $f$ in a metric space from of the continuity of $f$ in every compact subset of $E$

Question

Let $E$, $E'$ be two metric spaces,$f$ a mapping of $E$ into $E'$. Show that if the restriction of $f$ to any compact subspace of $E$ is continuous, then $f$ is continuous in E.

Answer 1

HINT: If $\langle x_n:n\in\Bbb N\rangle$ is a sequence in $E$ converging to some $x\in E$, then $\{x_n:n\in\Bbb N\}\cup\{x\}$ is a compact set.