Euler differential identity characterizes homogeneity
= Euler differential identity characterizes homogeneity
{c}
On an open domain invariant under positive scaling, $Df_x(x)=cf(x)$ implies $f(\lambda x)=\lambda^cf(x)$. Indeed, $\frac d{d\lambda}[\lambda^{-c}f(\lambda x)]=\lambda^{-c-1}[Df_{\lambda x}(\lambda x)-cf(\lambda x)]=0$. The converse follows by differentiating the scaling identity at $\lambda=1$. Connectedness of the whole domain is unnecessary: the argument works along each positive ray.