I was wondering if there exists a common notation for local maxima / peaks of some function $f: \mathbb{R}\rightarrow\mathbb{R}$. More specifically, I am looking for something similar to $$x'=\operatorname{arg\,\max_x}f(x)$$ but instead of one global maxima, I am interest in the set of all local maxima. I could imaging a notation similar to $$\{x'\}=\operatorname{arg\, local\max_x}f(x)$$ (I came up with that myself. Not very happy with it.)
Using this notation (stolen from Wikipedia) $${\displaystyle {\underset {x\in S}{\operatorname {arg\,max} }}\,f(x):=\{x\mid x\in S\wedge \forall y\in S:f(y)\leq f(x)\}.}$$ one could maybe construct something using this local maxima definition, also from Wikipedia.
However, this gets convoluted quite quickly. I am looking for something simple and clean. It doesn't have to be super exact, it just needs to convey the idea.
Thanks
Something like this?
arg local max $f(x)$ = $\{x | \exists r>0$ s.t. $\forall y \in (x-r,x+r), f(y) \leq f(x) \}$