Consider a function $f: \mathbb{R} \to \mathbb{R}$. As usual, $f$ is non-increasing if $f(x) \geq f(y)$ for all $x < y$. We also have the condition $f'(x) \leq 0$ $\forall x$, provided that $f$ is differentiable.
I can think of various generalizations of the notion of monotonicity:
- There exists a finite positive number $h > 0$ such that for all $x$, there exists $y$ such that $x \leq y \leq x + h$ and $f(x) \geq f(y)$.
- There exists a finite non-negative number $h \geq 0$ such that for all $x$, $f(x) \geq f(y)$ $\forall y \geq x + h$.
My questions are:
- Is there any generalization of monotonicity, similar to those above and has been studied in the literature? Any pointer to references would be much appreciated.
- If there is, then is there any condition in the line of the above derivative condition?