# 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?

S.L.

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Since $\phi$ is smooth iff each of the $n$ component functions of $\phi$ are smooth, you may as well assume $n=1$. (I wish I had more information to give you!) –  Jason DeVito Jan 3 '11 at 0:17
This is fantastic! I was about to ask this very question :D –  uncookedfalcon Dec 3 '12 at 4:23

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 According to the article and review, it had been an unpublished conjecture of Rådström.

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Thanks a lot for these links! –  Sam Jan 3 '11 at 1:52
@S.L.: You're welcome. Thank you for asking; I had fun finding the answer. –  Jonas Meyer Jan 3 '11 at 2:38
The old links to the article died. Currently the following work: digizeitschriften.de/en/dms/toc/?PPN=PPN35397434X_0020 for Math. Scand. 20 (1967), and digizeitschriften.de/en/dms/img/… for the article. (These may be added to the answer at some point if it seems worth it to "front page" the thread.) –  Jonas Meyer Mar 31 at 1:43