Prove that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on a maximum value on $[a,b]$.

Problem: 

Prove that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on a maximum value on $[a,b]$.

Answer: 

It is true that if $f(x)$ is continuous on $[a,b]$, then $f(x)$ takes on a maximum value on $[a,b]$.