Real function/Derivative/Monotonicity/Fact/Proof

Proof

(1). It is enough to prove the statements for increasing functions. If ${}f$ is increasing and ${}x\in I$, then the difference quotient fulfills

${}{\frac {f(x+h)-f(x)}{h}}\geq 0\,$

for every ${}h$ with ${}x+h\in I$. This estimate carries over to the limit as ${}h\rightarrow 0$, and this limit is ${}f'(x)$.
Suppose now that the derivative is ${}\geq 0$. We assume, in order to obtain a contradiction, that there exist two points ${}x<x'$ in ${}I$ with ${}f(x)>f(x')$. Due to the mean value theorem, there exists some ${}c$ with ${}x<c<x'$ and

${}f'(c)={\frac {f(x')-f(x)}{x'-x}}<0\,,$

which contradicts the condition.
(2). Suppose now that ${}f'(x)>0$ holds with finitely many exceptions. We assume that ${}f(x)=f(x')$ holds for two points ${}x<x'$. Since ${}f$ is increasing, due to the first part, it follows that ${}f$ is constant on the interval ${}[x,x']$. But then ${}f'=0$ on this interval, which contradicts the condition that ${}f'$ has only finitely many zeroes.

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