= Solution
For a <homogeneous function>, fix $x\in U$ and differentiate $f(\lambda x)=\lambda^cf(x)$ at $\lambda=1$. The <chain rule> gives
$$
\boxed{Df_x(x)=cf(x)}.
$$
Conversely, 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 gives
$$
h'(\lambda)=\lambda^{-c-1}\{Df_{\lambda x}(\lambda x)-cf(\lambda x)\}=0.
$$
Therefore $h$ is constant on the connected interval $(0,\infty)$ and equals $h(1)=f(x)$. Hence $\boxed{f(\lambda x)=\lambda^cf(x)}$ for every positive $\lambda$. This proves <Euler differential identity characterizes homogeneity>; no connectedness assumption on the whole domain is needed.
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