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In considering certain types of generalized smooth spaces, one may try to describe the smooth structure on a space by specifying the smooth curves. Boman's Theorem shows that this is sufficient to describe the smooth structure on a smooth manifold.
The following theorem appears (as Theorem 1) in the paper Boman 1967, where it is first proved, by Jan Boman:
Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^\infty(\mathbb{R},\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}, \mathbb{R}^d)$. Then $f \in C^\infty(\mathbb{R}^d, \mathbb{R})$.
Here, $\mathbb{R}^d$ is a Cartesian space, and $C^\infty(X,Y)$ is the set of smooth maps from $X$ to $Y$.
The theorem is quoted with a proof in Kriegl & Michor 1997 (Theorem 3.4).
This theorem is for smooth functions, that is $C^\infty$ maps. A similar theorem could be stated for continuous functions, that is $C^0$ maps. The situation is slightly less than ideal, however, for continuously differentiable functions, that is $C^1$ maps, or more generally $C^p$ maps for $0 \lt p \lt \infty$.
Boman 1967 has this as part of Theorem 2:
Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^p(\mathbb{R},\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}, \mathbb{R}^d)$. Then $f \in C^{p-1}(\mathbb{R}^d, \mathbb{R})$.
Note that $p$ has become $p - 1$ in the conclusion. (Boman's full Theorem 2 gives stronger results involving Lipschitz conditions.)
Boman's Theorem 8 gives the desired result if we use parametrized surfaces instead of curves:
Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^p(\mathbb{R}^2,\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}^2, \mathbb{R}^d)$. Then $f \in C^p(\mathbb{R}^d, \mathbb{R})$.
Here we have $\mathbb{R}^2$ instead of $\mathbb{R}$ as the domain of $u$.
Boman's Theorem 3 guarantees such counterexamples as
(continuously extended so that $f(0,0) = 0$). Given any smooth —or even $C^1$— curve $u\colon t \mapsto (g(t), h(t))$, it may be shown (by several tedious cases) that $(f \circ u)'$ is continuous. Nevertheless, $f$ is not $C^1$ at $(0,0)$. (The general pattern, expressed in Boman's Theorem 10, is to use a non-polynomial function that is homogeneous in degree $p$ and $C^p$ except at $\vec{0}$. So long as $d \gt 1$, such functions exist.) Additionally, $(f \circ u)'$ exists even if $u$ is merely differentiable, but $f$ is not even differentiable at $(0,0)$.
This does not contradict the well known theorem (often taken as a definition!) that a function is $C^1$ already if only its partial derivatives are continuous; while the partial derivatives of $f$ may be expressed as derivatives of $f \circ u$ for appropriate smooth $u$ (taken from a space of curves identifiable with $d \mathbb{R}^d$), the continuity of the partial derivatives requires not that $(f \circ u)'(t)$ be continuous in $t$ (although this will follow) but that it be continuous in $u$.
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
Last revised on April 10, 2016 at 22:47:50. See the history of this page for a list of all contributions to it.