Suppose $f$ is a function from a complete metric space $X$ to a metric space $Y$, and suppose $Y$ has points $y_{0}$, $y_{1}$ such that the subsets $f^{-1}(y_{0})$ and $f^{-1}(y_{1})$ are both dense in $Y$.
- Show that $f$ is not a pointwise limit of continuous functions $X \to Y$.
- Show that in the above situation, if $f$ maps every point of $X$ either to $y_{0}$ or to $y_{1}$ and if $f^{-1}(y_{1})$ is countable, then $f$ cannot even be expressed as a pointwise limit of functions $X \rightarrow Y$ that are continuous at all points of $f^{-1}(y_{1})$; but that it can be expressed as a pointwise limit of functions $X \rightarrow Y$ that are continuous at all points of $f^{-1}(y_{0})$.
Questions:
- I was wondering how to use Baire's theorem to prove the 1st bullet.
- How to prove the 2nd bullet? We can use the Dirichlet function to give a counterexample to the 1st part, but any direct proof?
Thank you.