Prove that $f$ is an increasing function if $f'(x)$ is more than zero for all real values for $x$ I'm having some difficulty with proving this theory. What I do know is how to prove that it is a constant function when $f'(x) = 0$ by simplying assuming that f is not constant and contradict the supposition. In this case, what should I do instead?

$f'(x) > 0$ everywhere means $f$ is continuous everywhere.

And the mean value theorem says that for any $a, b; a < b$ that there is a $c: a < c < b$ where $f'(c)=\frac {f(b)-f(a)}{b-a}$.

But we know $f'(c) > 0$ so $f(b) > f(a)$.