Maximal Domain of Continuity

Question

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).

Answer 1

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