Geodesic
Some properties of the modulus of continuity
Matematičeskie zametki, Tome 9 (1971) no. 5, pp. 495-500 Cet article a éte moissonné depuis la source Math-Net.Ru

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Necessary and sufficient conditions are derived for the continuity and semiadditivity of the modulus of continuity of a function $f(x)$ given on a compact $\Omega$ in $n$-dimensional euclidean space. 
@article{MZM_1971_9_5_a2,
     author = {I. M. Kolodii and F. Hil'debrand},
     title = {Some properties of the modulus of continuity},
     journal = {Matemati\v{c}eskie zametki},
     pages = {495--500},
     year = {1971},
     volume = {9},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1971_9_5_a2/}
}
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I. M. Kolodii; F. Hil'debrand. Some properties of the modulus of continuity. Matematičeskie zametki, Tome 9 (1971) no. 5, pp. 495-500. http://geodesic.mathdoc.fr/item/MZM_1971_9_5_a2/