Assume the following:
- $U \subseteq \mathbb{R}^m$ is a closed and connected set (not necessarily bounded),
- $\Xi \in Lip_{\alpha}(U,X)$, where $X \subseteq \mathbb{R}^n$,
- $g \in C^1(X,Y)$, where $X \subseteq \mathbb{Y}^m$.
Is the function $G = g \circ \Xi: U \rightarrow Y$ Lipschitz?
Up to now, I cannot find neither a way to prove it true, nor a counterexample to prove it wrong. The problem is that: $\Xi(U)$ is not necessarily bounded, so I cannot assume $g$ Lipschitz on $X$, even if $g \in C^1(X)$. If $g$ would be Lipschitz, then the problem is trivial.