# Maximal Domain of Continuity

Let $$f:(X,T)\rightarrow (Y,S)$$ be a function between Polish spaces. Does there necessarily exists a (non-trivial: ie not a point or the emptyset) open subset $$\tilde{X}\subseteq X$$, on which $$f|_{\tilde{X}}$$ becomes continuous.

Note: a maximum here (is not necessarily unique).

Of course there need not exist such an open set. Just consider $$X=Y=\Bbb R$$ (with standard topology) and $$f(x)=x+\mathbf 1_{\Bbb Q}(x).$$