Geodesic
Absolute upper semicontinuity
Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 395-399 
V. D. Ponomarev. Absolute upper semicontinuity. Matematičeskie zametki, Tome 22 (1977) no. 3, pp. 395-399. http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a8/
@article{MZM_1977_22_3_a8,
     author = {V. D. Ponomarev},
     title = {Absolute upper semicontinuity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {395--399},
     year = {1977},
     volume = {22},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1977_22_3_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $\varphi(t'')-\varphi(t')\le\int_{t'}^{t''}g(s)\,ds$.