Euler differential identity characterizes homogeneity

On an open domain invariant under positive scaling, $D{f}_x(x)=cf(x)$ implies $f(\lambda x)={\lambda }^{c}f(x)$. Indeed, $\frac{d}{d\lambda }[{\lambda }^{-c}f(\lambda x)]={\lambda }^{-c-1}[D{f}_{\lambda x}(\lambda x)-cf(\lambda x)]=0$. 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.