Problem 2-35 on page 34 of Spivak's *Calculus on Manifolds* states 

> 
> If $f: \mathbb{R}^n: \to \mathbb{R}$ is differentiable and $f(0) =0$,
>  prove that there exist $g\_i: \mathbb{R}^n \to \mathbb{R}$ such that
>  $f(x) = \sum\_{i=1}^n x^ig\_i(x)$. 
> 
> 
> Hint: if $h\_x(t) = f(tx)$, then $f(x)
>  = \int\_0^1 h\_x'(t)$.
> 
> 
> 

I understand the answer he wants, but hasn't he left out a hypothesis that would ensure $h\_x'(t)$ is integrable? (For instance the continuity of $df$.) 

Secondly, I'm wondering if this theorem is used anywhere. Perhaps in differential geometry it might be useful to write locally $f \in C^\infty (M)$ as $\sum\_i g\_i(x) x\_i$ for some coordinates $x\_i$.