second answer:
under constructionSuppose we add some good conditions for $X$ and $Y$. Complete or Polish or something. Then it would suffice to prove this:
LEMMA. Let $X, Y$ satisfy (conditions to be determined). Let $D \subseteq X$ have the subspace topology. Suppose $f : D \to Y$ is continuous. Then there exists $g : X \to Y$ such that: (a) $g(x)=f(x)$ for all $x \in D$; (b) $g$ is continuous at each point of $D$.
That would be a result of point-set topology. And it would imply the result we want. Presumably topologists already know this result, together with the conditions to impose. For example, it should be easy to do the case $X=Y=[0,1]$.