Question: Let $A \subseteq \mathbb{R}$. Let $f:A \rightarrow \mathbb{R}$ be continuous function.

If $A$ is bounded, is $f(A)$ necessarily bounded?

If $A$ is closed, is $f(A)$ necessarily closed?

Does $f$ must be uniformly continuous to preserve those properties? and not only continuous? 
I manage to figure out that if $f$ uniformly continuous,then $f(A)$ is bounded.

However, I am stuck on the first part of the question.

Answer: 
**HINT**: Try an *unbounded* closed subset for $A$.
Let $f(x)=e^{-1/(x^{2}+1)}$ for $x\in\mathbb{R}$, $1$ is an accumulation point of $f(\mathbb{R})$ which does not belong to $f(\mathbb{R})$, so $f(\mathbb{R})$ is not closed.