Problem: Let $f:\mathbb{R} \to \mathbb{R} $ be a continuous function. Prove that image of a bounded set is bounded. Will the conclusion still holds if we take $f: A\subseteq \mathbb{R} \to \mathbb{R} $.
Solution: et $A$ be any bounded subset of $\mathbb{R} $, then there exists $M>0$ such that for all $a\in A$ we have $\vert a \vert \leq M$. Therefore, $A \subseteq [-M,M].$ Since a continuous function maps a compact set to a compact set and we have \[ A\subseteq B \implies f(A) \subseteq f(B), \] so, \[ f(A) \subseteq f([-M,M]), \] which implies $f(A)$ is bounded.
Note that if we do not assume the function to be defined on $\mathbb{R} $, this conclusion need not be true. For example, take $f(x) = \tan x$ where $x\in \left( \frac{-\pi}{2}, \frac{\pi}{2} \right) $.