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On level sets of typical continuous functions defined on metric locally compact spaces. (Russian) Zbl 0604.26002

Let E be a metric compact space without isolated points, let C(E) be the space of all real continuous functions on E with the supremum norm and let F(E) be the subspace of all \(f\in C(E)\) which have the following three properties: 1. f does not have any non-strict local extreme. 2. The set A of all local extrema of f is dense in E and f(A) is dense in F(E). 3. If a,b\(\in A\) are two non-equal points, then \(f(a)\neq f(b).\) Then F(E) is residual in C(E). The level sets \(S_{\alpha}\), for each \(\alpha\) and for each f of F(E), are of the form \(P_{\alpha}\) or \(P_{\alpha}\cup \{x_{\alpha}\},\) where \(P_{\alpha}\) is a non-dense subset of E and no point of \(P_{\alpha}\) is a local extreme of f and \(x_{\alpha}\) is a local extreme of f. Also, the subset \(W(E)=\{f\in F(E):\) for each \(\alpha\) and for each \(u\in P_{\alpha}\) each neighbourhood of u contains at least one component of \(E-P_{\alpha}\}\) of F(E) is residual in C(E).
Reviewer: L.Mišík

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C35 Function spaces in general topology