Are continuous functions on a closed interval nowhere infinite?

Question

My intuition says YES. Am I correct? The following is the proof of my intuition.

Let a function $f(x)$ be continuous in the interval $[a,b]$ and suppose it is infinite at a point $c$ in the interval.

$\lim\limits_{\Delta x \to0} f(c\ +\Delta x)\ -f(c)= \text{not defined}$

Therefore $f(x)$ is not continuous at $c$

This is a contradiction. Therefore our supposition is incorrect. Therefore $f(x)$ cannot be infinite at point $c$ in the interval. Since $c$ is an arbitrary point, $f(x)$ cannot be infinite at any point in the interval $[a,b]$.

Is my proof correct?

Answer 1

As noted in the comments, function being "infinite" dosent make sense, but function tending to infinity, or the limit of the function being infinity, does make sense. In this sense, consider $[0,\infty)$, which is closed and let the function be $e^x$, Then, as $x\to\infty$, the function also "tends to infinity".

On the other hand, if the interval were closed and bounded(or compact), your guess is right

Answer 2

A continuous real function on a closed finite interval is bounded there, and achieves its maximal and minimal values within the interval. These two are the well known Weierstrass Theorems I and II .

I suppose and hope the above answers your question...