Let X be any set and define $d: X \times X \to \Bbb R$ by
$d(x,y)= \begin{cases} 0 & x=y \\ 1 & x \neq y \end{cases}$.
Classify all continuous functions $f: X \to X$ using the discrete metric on both sets.
This is my first course in topology and I am struggling to make sense of how I would classify all of the continuous functions in this case. I can see that $f(x) \neq f(a)$, then there exists $\epsilon > 0$ which do not satisfy the definition of continuity, so is it just that every continuous function in this case must map all of the elements in the domain to exactly one value in the codomain? Or am I just completely misunderstanding the question?
Thanks!
Best Answer
HINT: Is there any function from $X$ to $X$ that is not continuous?
Your comment about what happens when $f(x)\ne f(a)$ suggests that you have some misunderstanding of continuity, because the identity function from $X$ to $X$ is always continuous, no matter what metric you’re using, and it’s never constant unless $X$ has only one point.