Let $d\in\mathbb N$, $U\subseteq\mathbb R^d$ be open, $v\in C^1(U,\mathbb R^d)$ and $K\subseteq U$ be compact.
Can we show that $\left.f\right|_K$ is Lipschitz continuous?
Since $f$ is differentiable, there is a $(\delta_x)_{x\in U}\subseteq(0,\infty)$ with $$\left\|f(x)-f(y)\right\|\le\underbrace{\left(\left\|{\rm D}f(x)\right\|_{\mathfrak L(\mathbb R^d)}+1\right)}_{=:\:c_x}\left\|x-y\right\|\;\;\;\text{for all }y\in B_{\delta_x}(x)\tag1.$$ And since $K$ is compact, there are $k\in\mathbb N$, $x_1,\ldots,x_k\in K$ with $$K\subseteq\bigcup_{i=1}^kB_{\delta_{x_i}}(x_i)\tag2.$$ I would like to conclude from that, but I guess it will not work.
