Undergrad 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&#039; of Y such that <br /> S = \overline{f^{-1}(\{c\})} \cap \overline{f^{-1}(\{c&#039;\})} \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&#039;\}) 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 <br /> \mathbb{R}^2 \to \mathbb{R} : (x,y) \mapsto \begin{cases} \frac{x^2 - y^2}{x^2+ y^2} &amp; (x,y) \neq (0,0) \\ \mbox{(any real number)} &amp; (x,y) = (0,0) \end{cases} in the limit (x,y) \to (0,0) or <br /> [0, \infty) \to \mathbb{R} : x \mapsto \begin{cases} \sin (x^{-1}) &amp; x \neq 0 \\<br /> \mbox{(any real number)} &amp; 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&#039; \in f^{-1}(\{c&#039;\}) having x as their common limit, so that <br /> \lim_{n \to \infty} f(x_n) = c \neq c&#039; = \lim_{n \to \infty} f(x_n&#039;) 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&#039; \in f^{-1}(\{c&#039;\}) having x as their common limit, so that <br /> \lim_{n \to \infty} f(x_n) = c \neq c&#039; = \lim_{n \to \infty} f(x_n&#039;) 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.