Geodesic
The degree of rational approximation of functions and their differentiability
Izvestiya. Mathematics , Tome 17 (1981) no. 3, pp. 595-600.

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Denote by $R_n(f,E)$ the least uniform deviation of the function $f(x_1,\dots,x_m)$, defined in a subset $E$ of $m$-dimensional Euclidean space, from the rational functions $R_n(x_1,\dots,x_m)$ of degree $\leqslant n$. It is shown that if $\sum R_n(f,E)\infty$, then, a.e. on $E$, $f(x_1,\dots,x_m)$ has a total differential. The case $m=1$ was previously treated by E. P Dolzhenko. Bibliography: 9 titles. 
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E. A. Sevast'yanov. The degree of rational approximation of functions and their differentiability. Izvestiya. Mathematics , Tome 17 (1981) no. 3, pp. 595-600. http://geodesic.mathdoc.fr/item/IM2_1981_17_3_a6/

[1] Gonchar A. A., “O nailuchshikh priblizheniyakh ratsionalnymi funktsiyami”, Dokl. AN SSSR, 100:2 (1955), 205–208 | Zbl

[2] Dolzhenko E. P., “Skorost priblizheniya ratsionalnymi drobyami i svoistva funktsii”, Matem. sb., 56 (1962), 403–432 | Zbl

[3] Dolzhenko E. P., “Nekotorye tochnye integralnye otsenki proizvodnykh ratsionalnykh i algebraicheskikh funktsii. Prilozheniya”, Anal. Math., 4 (1978), 247–268 | DOI | MR

[4] Dolzhenko E. P., “O svyazi mezhdu svoistvami funktsii i skorostyu ikh priblizheniya ratsionalnymi funktsiyami so svobodnymi polyusami i polinomami”, Problemy razvitiya prikladnykh matematicheskikh issledovanii, Tezisy dokladov, ch. 2 (IV respublikanskaya konferentsiya matematikov Belorussii), Belorusskii gos. universitet, Minsk, 1975, 112–120

[5] Dolzhenko E. P., “O svoistvakh funktsii neskolkikh peremennykh, dostatochno khorosho priblizhaemykh ratsionalnymi drobyami”, Izv. AN SSSR. Ser. matem., 26 (1962), 641–652 | Zbl

[6] Saks S., Teoriya integrala, IL, M., 1949

[7] Petrovskii I. G., Oleinik O. A., “O topologii deistvitelnykh algebraicheskikh poverkhnostei”, Izv. AN SSSR. Ser. matem., 13 (1949), 389–402 | MR | Zbl

[8] Dolzhenko E. P., “Otsenki proizvodnykh ratsionalnykh funktsii”, Izv. AN SSSR. Ser. matem., 27 (1963), 9–28

[9] Sevastyanov E. A., “Ratsionalnaya approksimatsiya i absolyutnaya skhodimost ryadov Fure”, Matem. sb., 107(149):2(10) (1978), 227–244 | MR