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

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

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

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

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

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