What’s the name (or reference) for the real-analytic lemma
A $\mathrm C^\infty$ function $f:\mathbf R\to\mathbf R$ with $f(0) = 0$ is $f(x) = xg(x)$ with $g$ also $\mathrm C^\infty\,$
and the multidimensional variant
A $\mathrm C^\infty$ function $f:\mathbf R^n\to\mathbf R$ with $f(0) = 0$ is $f(x) =\sum_i x_ig_i(x)$ with the $g_i$ also $\mathrm C^\infty\,$?
I can easily prove at least the one-dimensional variant using Taylor’s theorem (and actually need the Schwartz function variant, anyway), but I could swear I saw it named somewhere, as an ingredient for either Borel’s theorem on Taylor series or Whitney’s extension theorem.