On the set of local extrema of a subanalytic function

Abstract

Let 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 F(M) the family of the subsets of M that belong to the category F. Let f:X→R be a subanalytic function on a subset X∈F(M) such that the inverse image under f of each interval of R belongs to F(M). Let Max(f) be the set of local maxima of f and consider its level sets Maxλ(f):=Max(f)∩{f=λ}={f=λ}∖Cl({f>λ}) for each λ∈R. In this work we show that if f is continuous, then Max(f)=⨆λ∈RMaxλ(f)∈F(M) if and only if the family {Maxλ(f)}λ∈R is locally finite in M. If we erase continuity condition, there exist subanalytic functions f:X→M such that Max(f)∈F(M), but the family {Maxλ(f)}λ∈R is not locally finite in M or such that Max(f) is connected but it is not even subanalytic. We show in addition that if F is the category of subanalytic sets and f:X→R is a (non-necessarily continuous) subanalytic map f that maps relatively compact subsets of M contained in X to bounded subsets of R, then Max(f)∈F(M) and the family {Maxλ(f)}λ∈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 F is either the category of semianalytic sets or the category of C-semianalytic sets and f is the restriction to an F-subset of M of an analytic function on M, then the family {Maxλ(f)}λ∈R is locally finite in M and Max(f)=⨆λ∈RMaxλ(f)∈F(M). We also show that if the category F contains the intersections of algebraic sets with real analytic submanifolds and X∈F(M) is not closed in M, then there exists a continuous subanalytic function f:X→R with graph belonging to F(M×R) such that inverse images under f of the intervals of R belong to F(M) but Max(f) does not belong to F(M). As subanalytic sets are locally connected, the set of non-openness points of a continuous subanalytic function f:X→R coincides with the set of local extrema Extr(f):=Max(f)∪Min(f). This means that if f:X→R is a continuous subanalytic function defined on a closed set X∈F(M) such that the inverse image under f of each interval of R belongs to F(M), then the set Op(f) of openness points of f belongs to F(M). Again the closedness of X in M is crucial to guarantee that Op(f) belongs to F(M). The type of results stated above are straightforward if F is an o-minimal structure of subanalytic sets. However, the proof of the previous results requires further work for a category F of subanalytic sets that does not constitute an o-minimal structure.