A non-empty intersection of closures of level sets implies discontinuity

[pasmith]

Let $X$ and $Y$ be topological spaces, and suppose $f: X \to Y$ is such that there exist distinct points $c$ and $c'$ of $Y$ such that $$S = \overline{f^{-1}(\{c\})} \cap \overline{f^{-1}(\{c'\})} \neq \varnothing.$$ What conditions must be placed on $X$ and $Y$ so that it follows that $f$ is discontinuous at each point of $S$? (Note that $f^{-1}(\{c\}) \cap f^{-1}(\{c'\})$ is necessarily empty: a function cannot take more than one value at any point of its domain.)

The statement appears to hold if $X$ and $Y$ are (locally) Euclidean, as for example $$\mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto \begin{cases} \frac{x^2 - y^2}{x^2+ y^2} & (x,y) \neq (0,0) \\ \mbox{(any real number)} & (x,y) = (0,0) \end{cases}$$ in the limit $(x,y) \to (0,0)$ or $$[0, \infty) \to \mathbb{R} : x \mapsto \begin{cases} \sin (x^{-1}) & x \neq 0 \\ \mbox{(any real number)} & x = 0 \end{cases}$$ in the limit $x \to 0$, but does it hold between more general spaces?

We do have the following constraints:

• If $X$ is discrete, then $f$ is necessarily continuous.
• If $Y$ is indiscrete, then $f$ is necessarily continuous.

My idea is that for each $x \in S$ we can construct two distinct sequences, $x_n \in f^{-1}(\{c\})$ and $x_n' \in f^{-1}(\{c'\})$ having $x$ as their common limit, so that $$\lim_{n \to \infty} f(x_n) = c \neq c' = \lim_{n \to \infty} f(x_n')$$ and $f$ is discontinuous at $x$. But this assumes that limits in $X$ and $Y$ are unique.

 

[andrewkirk]

My idea is that for each $x \in S$ we can construct two distinct sequences, $x_n \in f^{-1}(\{c\})$ and $x_n' \in f^{-1}(\{c'\})$ having $x$ as their common limit, so that $$\lim_{n \to \infty} f(x_n) = c \neq c' = \lim_{n \to \infty} f(x_n')$$ and $f$ is discontinuous at $x$. But this assumes that limits in $X$ and $Y$ are unique.

That works. It points us towards requiring the Hausdorff condition for space $Y$, which is perhaps the most commonly required condition in topology - so likely what the examiner was aiming for.
I wonder whether the weaker condition of preregularity would suffice. To investigate that we'd need to consider what happens in a preregular, non-Hausdorff space where the above points $c$ and $c'$ are not topologically distinguishable.