Open and closed sets and continuous functions

Question

Suppose that $(X,ρ)$ is a metric space, $f:(X,ρ)\rightarrow R$ a continuous function and $D$ a dense subset of $X$ so as $f(D)$ finite. Prove that:

(i) The range $f(X)$ is finite

(ii) For every $t\in f(X)$, $f^{-1}\ (\{t\})$ is simultaneously open and closed in $X$.

Answer 1

For (i), you should use the density of $D$ to prove that $f(D)=f(X)$. And (ii) is just a consequence from the fact that in a finite subset of $\mathbb R$, singletons are both open and closed.

Answer 2

$f[D]$ is dense in $f[X]$ by continuity. In any metric space (so in $\Bbb R$ too) a finite set is closed, so equals its own closure. Hence $f[X]=f[D]$ is finite.

Also, in a finite subset all singletons are open and closed and thus so are their inverse images, again by continuity.