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Absolute upper semicontinuity

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Abstract

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ t″ t′ g (s) ds.

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Authors and Affiliations

  1. Latvian State University, Latvia

    V. D. Ponomarev

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  1. V. D. Ponomarev
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Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.

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Ponomarev, V.D. Absolute upper semicontinuity. Mathematical Notes of the Academy of Sciences of the USSR 22, 711–713 (1977). https://doi.org/10.1007/BF02412500

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  • DOI: https://doi.org/10.1007/BF02412500

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