If $f$ is continuous on a closed interval $[a, b]$, then $f$ attains both an absolute maximum value $M$ and an absolute minimum value $m$ in $[a, b]$. That is, there are numbers $x_{1}$ and $x_{2}$ in $[a, b]$ with $f\left(x_{1}\right)=m, f\left(x_{2}\right)=M$, and $m \leq f(x) \leq M$ for every other $x$ in $[a, b]$ Proved later in [[Real Analysis]]