This was proved by Jan Boman in the paper "Differentiability of a function and of its compositions with functions of one variable", Math. Scand. 20 (1967), 249-268. (The theorem as stated is for the case $n=1$, but that is no problem as Jason DeVito already mentioned in a comment.) Here's an online version, and here's the MathSciNet link. According to the article and review, it had been an unpublished conjecture of Rådström.