Absolute upper semicontinuity

V. D. Ponomarev

Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 395-399.

Résumé

It is proved that the following conditions are equivalent: the function $\varphi[a,b]\to R$ is absolutely upper semicontinuous (see [1]); $\varphi$ is a function of bounded variation with decreasing singular part; there exists a summable function $g:[a,b]\to R$ such that for any $t'\in[a,b]$ and $t''\in[t',b]$, we have $\varphi(t'')-\varphi(t')\le\int_{t'}^{t''}g(s)\,ds$.