If $f$ is a strictly increasing function, shouldn't $f'$ be always positive and never zero? Apparently there's this situation where the derivative can be $0$ if it's only at discrete points and not an interval.
How is that possible, when strictly increasing functions are, by definition, never of a zero slope?
Best Answer
"strictly increasing functions are, by definition, never of a zero slope": that is not the definition of a strictly increasing function. The correct one says
$$x_0<x_1\implies f(x_0)<f(x_1).$$
Geometrically, this can be expressed as "the slope of any chord is positive", but not as "the slope of any tangent is positive".