Is $A$ compact, $f(A)$ uniformly continuous and is $f^{-1}$ continuous?

Question

$X$ and $Y$ are metric spaces, $A\subseteq X$, $A$ is bounded. map $f:X\to Y$ is continuous.

Questions:

1. Is $A$ necessarily compact?
2. Is $f(A)$ uniformly continuous?
3. If given that $f$ is a bijection and $f(A)$ is compact, is $f^{-1}$ necessarily continuous?

Tried: If $A$ is closed and bounded, then $A$ must be compact hence compactness of $f(A)$ follows. So the question reduces to "is boundedness of $A$ sufficient for compactness?" Also, I think 3 is correct because bijection and compact domain and codomain should be enough for homeomorphism...

I am new to topology so really appreciate any help!

Answer 1

For 1,2 no. Consider $f(x)=\frac{1}{x}$ on $(0,1]$

For 3, no. Consider identity map from $[0,1]$ to $[0,1]$ , where the first one with discrete metric and the second one with standard metric.