real analysis - Existence of maximum value of a discontinuous function

If $f(x)$ takes a finite real value for all $x$ on the closed interval $[a,b]$, must there be a real number $M$ such that $M\geq f(x)$ for all $x$ on this interval? It seems that if not, there must be a point $c\in[a,b]$ such that $\lim_{x\to c}f(x)=+\infty$, and so $f(x)$ must be undefined at some point on this interval, but I don't know how to make this rigorous.

Edit: I see that $f(0)=0$, $f(x)=1/x$ on $(0,1]$ is a counterexample. I also see that I have been imprecise with terminology. Let me modify the question: Is there always a sub-interval $[a',b']$ with $a