# Relaxing Monotonicity of a Function $f:\mathbb{Z}\rightarrow \mathbb{R}$

Suppose a function $f:\mathbb{Z_+}\rightarrow \mathbb{R}$ fails monotonicity, but not by much.

For example $f(2)= .3$ and $f(z)=1/z$ otherwise. Here there exists a single point where the function is monotonic on either side.

We could also imagine a function $g(z)$ where $g(5)=.26$ and $g(z)=f(z)$ otherwise. Here, there is no point where the function is monotonic on either side, but there are finitely many points that fail monotonicity.

Are there names for any of these concepts or is it best to simply describe why these kinds of functions fail monotonicity?

• In this case you could say it is "eventually monotonic" (when $z$ is large). – Seth Dec 13 '14 at 17:11
• Alternatively, if you are familiar with the concept of measure you could say it is "almost everywhere" monotonic, i.e., the set on which it is not monotonic is of measure zero. In the standard lebesgue measure, any countable finite or countably infinite set is of measure zero. – JMoravitz Dec 13 '14 at 17:21
• Thank you, almost everywhere occurred to me, but I wanted to check if there were any specific names. But I suppose not. – Pburg Dec 13 '14 at 17:39

A function that has exactly one minimum/maximum (or turning point, if we're speaking of discrete functions) and is otherwise monotonic is called unimodal. If we have two maxima which are also turning points, such as in $1 ,3, 2, 3, 1$, such a function could be called bimodal. This can clearly be generalized to arbitrarily many turning points.