Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!


share|cite|improve this question
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
up vote 6 down vote accepted

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 According to the article and review, it had been an unpublished conjecture of Rådström.

share|cite|improve this answer
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: for Math. Scand. 20 (1967), and… 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 '15 at 1:43

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.