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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 <br />
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 <br /> \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 <br /> [0, \infty) \to \mathbb{R} : x \mapsto \begin{cases} \sin (x^{-1}) & x \neq 0 \\<br /> \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:
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 <br /> \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.
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} & (x,y) \neq (0,0) \\ \mbox{(any real number)} & (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}) & x \neq 0 \\<br /> \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 <br /> \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.