Absolute upper semicontinuity

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