Hadamard's lemma

Hadamard's lemma[1] — Let ${\displaystyle f}$ be a smooth, real-valued function defined on an open, star-convex neighborhood ${\displaystyle U}$ of a point ${\displaystyle a}$ in ${\displaystyle n}$-dimensional Euclidean space. Then ${\displaystyle f(x)}$ can be expressed, for all ${\displaystyle x\in U,}$ in the form:

$${\displaystyle f(x)=f(a)+\sum _{i=1}^{n}\left(x_{i}-a_{i}\right)g_{i}(x),}$$

where each ${\displaystyle g_{i}}$ is a smooth function on ${\displaystyle U,}$ ${\displaystyle a=\left(a_{1},\ldots ,a_{n}\right),}$ and ${\displaystyle x=\left(x_{1},\ldots ,x_{n}\right).}$

Corollary[1] — If ${\displaystyle f:\mathbb {R} \to \mathbb {R} }$ is smooth and ${\displaystyle f(0)=0}$ then ${\displaystyle f(x)/x}$ is a smooth function on ${\displaystyle \mathbb {R} .}$ Explicitly, this conclusion means that the function ${\displaystyle \mathbb {R} \to \mathbb {R} }$ that sends ${\displaystyle x\in \mathbb {R} }$ to

$${\displaystyle {\begin{cases}f(x)/x&{\text{ if }}x\neq 0\\\lim _{t\to 0}f(t)/t&{\text{ if }}x=0\\\end{cases}}}$$

is a well-defined smooth function on ${\displaystyle \mathbb {R} .}$

Corollary[1] — If ${\displaystyle y,z\in \mathbb {R} ^{n}}$ are distinct points and ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ is a smooth function that satisfies ${\displaystyle f(z)=0=f(y)}$ then there exist smooth functions ${\displaystyle g_{i},h_{i}\in C^{\infty }\left(\mathbb {R} ^{n}\right)}$ (${\displaystyle i=1,\ldots ,3n-2}$) satisfying ${\displaystyle g_{i}(z)=0=h_{i}(y)}$ for every ${\displaystyle i}$ such that

$${\displaystyle f=\sum _{i}^{}g_{i}h_{i}.}$$