I'm familiar with the proof that if the first derivative is always positive then the function is increasing which uses the Mean Value Theorem. How can it be shown that the function not being differentiable at some point does not affect this property? For example a function may have a jump discontinuity with the derivative not being defined at a point but still be increasing.
Can you prove that if a function’s derivative is positive for all points where the function is differentiable then it is increasing
calculusderivativesmonotone-functionsreal-analysis
Best Answer
It may not always be the case. Consider the "fractional part" function:
$$f(x) := \{x\} := x - \lfloor x \rfloor$$
This function, in some sense, returns the "decimal part" of $x$. Hence
...though, for negative numbers, it returns the decimal part in the "other direction":
...and so on.
The graph itself looks like $y=x$ for $0 \le x < 1$, repeated over and over again:
Where it is differentiable, $f'$ is always positive, but clearly it is not always increasing (because of the times it jumps down).