I have seen the following result $f :\mathbb R \to \mathbb R$ be a continuous function. Then $f(A)$ is bounded for all bounded subsets $A$ of $\Bbb R$. How to prove it?
Also in above result, if we replace the word "bounded" by closed then is it a result? If yes, then how to prove it ?
If $A$ is bounded in $\Bbb R$, then $\overline A$ is compact, therefore $ f\left[\overline A\right]$ is compact. Since compact sets are bounded and $f\left[\overline A\right]\supseteq f[A]$, the claim is proved.
Consider the map $\arctan:\Bbb R\to\Bbb R$, the image if which is $(-\frac\pi2,\frac\pi2)$ for the second.