Let $K \subset \mathbb R^d$ be a compact. Let $\phi_{\varepsilon} \colon K \rightarrow \mathbb R$ be continuous and converge uniformly to $\phi$. Suppose further that $\phi$ is Lipschitz continuous. Can we deduce that $\phi_{\varepsilon}$ are Lipschitz continuous for $\varepsilon$ small?
Thanks.