Teorema do Limite

Statement of the theorem

Thm: If $f$ is continuous on a closed and bounded interval $[a,b]$ then it is bounded on the interval, i.e. there exists some real number $N$ such that $|f(x)| \leq N \ \forall x \in [a,b]$.

Assumptions

The theorem assumes that $f$ is continuous on a closed and bounded interval (for example, not the real number line). Both of these assumptions are necessary. For example, consider $f(x)= \frac{1}{x}$ on $(0,1]$ - we have a continuous function on a bounded interval, but it manages to escape to infinity as $0$ is not there to keep it in check.