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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:

  1. 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 $f(x) \geq f(y)$.
  2. 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:

  1. 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.
  2. If there is, then is there any condition in the line of the above derivative condition?

2 Answers 2