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Discontinuity points of exactly \(k\)-to-one functions. (English) Zbl 1065.26003

Summary: For a natural number \(k\geq 1\) and a topological space \(X\) the following question is considered. If \(F\subseteq X\) is an infinite \(F_\sigma\)-set that contains no point isolated in \(X\), does there exist an exactly \(k\)-to-one function \(f:X@>\text{onto}>>X\) whose set of all discontinuity points is \(F\)? The answer is given for \(k=1\) if \(X\) is a separable metrizable space, and for \(k\geq 1\) if \(X=[0,1]\).

MSC:

26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C10 Special maps on topological spaces (open, closed, perfect, etc.)
54C30 Real-valued functions in general topology
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