Boundedness Theorem

If $f$ is Continuous on $[a,b]$ then $f$ is Bounded on $[a,b]$.

Formal Definition

Using the definitions of cont and bounded, Cont $⟹$ Bounded ($\forall ϵ>0,\exists \delta >0$ s.t $|x-c|<\delta ⟹|f(x)-f(c)|<ϵ$) $⟹$ ($\exists M>0$ s.t $\forall x\in [a,b],|f(x)|\le M$)