Let $f$ be continuous on $[a,b]$ where $a<b$ such that $f(x)\ne0$ for all $x\in[a,b]$. Prove that there is a $c>0$ where either $f(x)>c$ for all $x\in[a,b]$ or $f(x)<-c$ for all $x\in[a,b]$.
Is this about local/absolute minimum and maximum? I do not know how to approach this question and any guidance to at least start the approach would be very much appreciated. Thank you!