Artificial intelligent assistant

function is smooth iff the composition with any smooth curve is again smooth I'm stuck on the following part of a proof: Let $\phi: \mathbb R^m \to \mathbb R^n$ be a function such that $\gamma'(t) := \phi(\gamma(t))$ is smooth for every smooth function $\gamma: \mathbb R \to \mathbb R^m$. I want to show that $\phi$ is smooth under these assumptions. Could someone give me a pointer? Thanks in advance! S.L.

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 According to the article and review, it had been an unpublished conjecture of Rådström.

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