3.II.13B

Let $f$ be a real-valued differentiable function on an open subset $U$ of $\mathbf{R}^{n}$. Assume that $0 \notin U$ and that for all $x \in U$ and $\lambda>0, \lambda x$ is also in $U$. Suppose that $f$ is homogeneous of degree $c \in \mathbf{R}$, in the sense that $f(\lambda x)=\lambda^{c} f(x)$ for all $x \in U$ and $\lambda>0$. By means of the Chain Rule or otherwise, show that

$\left.D f\right|_{x}(x)=c f(x)$

for all $x \in U$. (Here $\left.D f\right|_{x}$ denotes the derivative of $f$ at $x$, viewed as a linear map $\mathbf{R}^{n} \rightarrow \mathbf{R}$.)

Conversely, show that any differentiable function $f$ on $U$ with $\left.D f\right|_{x}(x)=c f(x)$ for all $x \in U$ must be homogeneous of degree $c$.

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