For my research, I need to verify whether a function $f: [a, b] \to \mathbb{R}$ with $f(b) > f(a)$has a nondecreasing interval $(c, d) \subset [a, b]$ such that $f$ is increasing on that interval and $f(c) \geq f(x)$ for all $x \in [a, c]$. Here is an example where two of the intervals I'm looking for are highlitghted in green:
I can impose "nice" properties to my function if needed. It seems reasonable that at least absolute continuity is required, to discart functions like the Cantor function.
The statement makes sense to me, and I am almost sure it is true. There are strange functions, like $f(x) = x \sin(\frac{1}{x}) + x$, which cross $0$ at $x = 0$, but you cannot say that there is a neighborhood of $0$ where $f$ is nondecreasing. However, I am not looking for a neighborhood of any particular point.
There might be an simple solution to this, but I am fairly new to this concepts, and I have been stuck for a couple of days.
Thank you in advance.