If $f:X\to Y$ is continuous and $X$ is a totally bounded metric space, is $f(X)$ also bounded?

Question

If $X$ and $Y$ are metric spaces and $X$ is totally bounded and $f:X\to Y$ is continuous (not necessarily uniform), is it true that $f(X)$ is also bounded? How can we prove it or is there any counter example?

I tried:

$f(x)=\tan x$ for $\pi/2<x<\pi/2$, then $X$ is bounded but $f(X)$ is unbounded.

Is my counterexample correct? If not, is there any better counterexample? Or maybe is $f(X)$ bounded and how can we prove it?

Thanks!

Answer 1

That is correct. $1/x$ on $(0,1)$ is easier.