# Generalization of monotonicity and condition

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?
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Of course your condition 1 holds for all functions because you allow $y=x$ and $f(x)=f(y)$. With the much stronger condition

For all $x$ and all $h>0$ there exists $y$ with $x<y<x+h$ and $f(x)\ge f(y)$

even such wild functions as $$f(x)=\begin{cases}e^x&x\in\mathbb Q\\0&x\notin\mathbb Q\end{cases}$$ would be "pseudo-monotonic non-increasing". Therefore I doubt that this kind of property is a useful concept / has been investigated in the literature.

Your second idea, however, seems quite useful as it can describe something like "noisy" decreasing functions.

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I saw Property 2 used under the name of $\varepsilon$-monotonicity, where $\varepsilon$ is what you denoted by $h$. It appears in Caffarelli's work on the regularity of free boundaries, specifically in his paper A Harnack inequality approach to the regularity of free boundaries. II. Flat free boundaries are Lipschitz in Comm. Pure Appl. Math. 42 (1989), no. 1, 55–78. The book by Caffarelli and Salsa, A Geometric Approach to Free Boundary Problems, treats it on pages 205-207.

Be warned that the usage of various "[letter]-monotone" concepts is not at all consistent in the literature.

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