nLab
Boman's theorem

In the paper Differentiability of a function and of its compositions with functions of one variable, Jan Boman proves the following theorem:

Theorem

Let $f$ be a function from ${ℝ}^d$ to $ℝ$, and assume that the composed function $f\circ u$ belongs to $C^{\mathrm{\infty }}(ℝ,ℝ)$ for every $u\in C^{\mathrm{\infty }}(ℝ,{ℝ}^d)$. Then $f\in C^{\mathrm{\infty }}({ℝ}^d,ℝ)$.

The theorem is quoted with a proof in The convenient setting of global analysis by Kriegl and Michor (theorem 3.4).

References

• Jan Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 1967 249–268, MR237728 pdf

• Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. x+618 pp. ISBN: 0-8218-0780-3 html MR1471480