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$$\omega$$-primitives on $$\sigma$$-discrete metric spaces. (English) Zbl 1025.26003
Summary: This paper continues the investigation started in [Z. Duszyński, Z. Grande and S. P. Ponomarev, “On the $$\omega$$-primitive”, Math. Slovaca 51, 469-476 (2001; Zbl 0987.54031)], [J. Ewert and S. P. Ponomarev, “Oscillation and $$\omega$$-primitives”, Real Anal. Exch. 26, 687-702 (2001; Zbl 1025.26002), preceding review], [P. Kostyrko, “Some properties of oscillation”, Math. Slovaca 30, 157-162 (1980; Zbl 0441.54004)], in the case of $$\sigma$$-discrete metric spaces. It is shown that given an upper semicontinuous function $$f\:X\to [0,\infty ]$$, where $$X$$ is a $$\sigma$$-discrete dense in itself metric space, there exists a function $$F\:X\to \mathbb R$$ (called an $$\omega$$-primitive for $$f$$) whose oscillation equals $$f$$.

##### 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