The degree of rational approximation of functions and their differentiability

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.
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