 Solution

For a homogeneous function, fix $x\in U$ and differentiate $f(\lambda x)={\lambda }^{c}f(x)$ at $\lambda =1$. The chain rule givesConversely, assume this derivative identity throughout $U$. Along a positive ray define $h(\lambda )={\lambda }^{-c}f(\lambda x)$. The ray stays in $U$ by hypothesis, and linearity of the derivative givesTherefore $h$ is constant on the connected interval $(0,\infty )$ and equals $h(1)=f(x)$. Hence $\boxed{f(\lambda x)={\lambda }^{c}f(x)}$ for every positive $\lambda$. This proves Euler differential identity characterizes homogeneity; no connectedness assumption on the whole domain is needed.