Assume that $(t,x)$ $\in$ $D \subseteq \mathbb{R}\times \mathbb{R}$ $^n$ non-empty, open, and connected set, and that $f:D\longrightarrow \mathbb{R}^n$.
Prove that $C^1(D)\subseteq$ $\text{locallyLip}_x(D)$ $\subseteq C(D)$ with respect to $x$.
The definition of $\text{locallyLip}_x(D)$ is as follows:
$f(t,x)$ is locally Lipschitz on $D$ with respect to $x$, if for every $(t,x)\in D$, there exists an open set $U\in D$ with $(t,x)\in U$, such that $f$ is uniformly Lipschitz on $U$, with respect to $x$ on $D$.
I am only used to applying Lipschitz condition in the context of analysis or differential equations, and thus I am not sure I understand how to show these inclusions.