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On the existence of \(\omega \)-primitives on arbitrary metric spaces. (English) Zbl 1053.26003
Summary: In this paper a final solution to the problem of the existence of \(\omega \)-primitives on an arbitrary metric space \((X,d)\) is given. Namely, it is shown that if \(f:X\to 0\) is an upper semicontinuous function, vanishing at each isolated point of \(X\), then there exists a function \(F:X\to \mathbb R\) whose oscillation equals \(f\) at each point of \(X\). We call such a function \(F\) an \(\omega \)-primitive for \(f\). Moreover, an \(\omega \)-primitive can always be found in at most Baire class 2.

MSC:
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
54C30 Real-valued functions in general topology
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References:
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