Hadamard’s lemma

Hadamard’s lemma: Any smooth function $f$ in a starlike neighborhood of a point $z$ is representable in the form

$$f(x)=f(z) \sum_{i=1}^n

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(x_i-z_i)g_i(x), \;g_i\in {C^{\infty}(U)},$$
where $g_i$ are smooth functions.