Can non-smooth function (first derivative is not defined at some points) be monotonic (including points where first derivative does not exist)?

Question

Rule that "for a function that has its first derivative on an interval" the function is monotonic if $f'(x) >= 0$ / $f'(x) < 0$ etc.

But what if the function does not have its first derivative everywhere on an interval (it is not defined in separate points or on the entire interval) - can the function still be monotonic on that interval and how to detect / prove that ?

Answer 1

Yes, that does exist. Just take two linear functions with different positive slope and combine those at any point. There will be a kink at that point and hence not differentiable there. The general definition is that if $x \leq y$ then $f(x) \leq f(y)$ for $x,y$ in the domain.