Uniform convergence of continuous functions with Lipschitz limit 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.

For $d=1$ and $K=[0,1]$ take $\phi _\epsilon =\epsilon \sqrt x$
Then $\phi _ {\epsilon } $ converges uniformly to the constant function $f\equiv 0$
which is a lipschitz function
But no $\phi _ \epsilon $ is lipschitz.
