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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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