On the set of local extrema of a subanalytic function

Abstract

Let $${\mathfrak {F}}$$ be a category of subanalytic subsets of real analytic manifolds that is closed under basic set-theoretical operations (locally finite unions, difference and product) and basic topological operations (taking connected components and closures). Let M be a real analytic manifold and denote $${\mathfrak {F}}(M)$$ the family of the subsets of M that belong to the category $${\mathfrak {F}}$$. Let $$f:X\rightarrow \mathbb {R}$$ be a subanalytic function on a subset $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$. Let $$\mathrm{Max}(f)$$ be the set of local maxima of f and consider its level sets $$\mathrm{Max}_\lambda (f):=\mathrm{Max}(f)\cap \{f=\lambda \}=\{f=\lambda \}{\setminus }{\text {Cl}}(\{f>\lambda \})$$ for each $$\lambda \in \mathbb {R}$$. In this work we show that if f is continuous, then $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$ if and only if the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. If we erase continuity condition, there exist subanalytic functions $$f:X\rightarrow M$$ such that $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$, but the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is not locally finite in M or such that $$\mathrm{Max}(f)$$ is connected but it is not even subanalytic. We show in addition that if $${\mathfrak {F}}$$ is the category of subanalytic sets and $$f:X\rightarrow \mathbb {R}$$ is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of $$\mathbb {R}$$, then $$\mathrm{Max}(f)\in {\mathfrak {F}}(M)$$ and the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M. An example of this type of functions are continuous subanalytic functions on closed subanalytic subsets of M. The previous results imply that if $${\mathfrak {F}}$$ is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an $${\mathfrak {F}}$$-subset of M of an analytic function on M, then the family $$\{\mathrm{Max}_\lambda (f)\}_{\lambda \in \mathbb {R}}$$ is locally finite in M and $$\mathrm{Max}(f)=\bigsqcup _{\lambda \in \mathbb {R}}\mathrm{Max}_\lambda (f)\in {\mathfrak {F}}(M)$$. We also show that if the category $${\mathfrak {F}}$$ contains the intersections of algebraic sets with real analytic submanifolds and $$X\in {\mathfrak {F}}(M)$$ is not closed in M, then there exists a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ with graph belonging to $${\mathfrak {F}}(M\times \mathbb {R})$$ such that inverse images under f of the intervals of $$\mathbb {R}$$ belong to $${\mathfrak {F}}(M)$$ but $$\mathrm{Max}(f)$$ does not belong to $${\mathfrak {F}}(M)$$. As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function $$f:X\rightarrow \mathbb {R}$$ coincides with the set of local extrema $$\mathrm{Extr}(f):=\mathrm{Max}(f)\cup \mathrm{Min}(f)$$. This means that if $$f:X\rightarrow \mathbb {R}$$ is a continuous subanalytic function defined on a closed set $$X\in {\mathfrak {F}}(M)$$ such that the inverse image under f of each interval of $$\mathbb {R}$$ belongs to $${\mathfrak {F}}(M)$$, then the set $$\mathrm{Op}(f)$$ of openness points of f belongs to $${\mathfrak {F}}(M)$$. Again the closedness of X in M is crucial to guarantee that $$\mathrm{Op}(f)$$ belongs to $${\mathfrak {F}}(M)$$. The type of results stated above are straightforward if $${\mathfrak {F}}$$ is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category $${\mathfrak {F}}$$ of subanalytic sets that does not constitute an o-minimal structure.