What is locally Lipschitz?

$f$ is locally Lipschitz in $x$ if every point of $U$ has a neighborhood $V$ and a constant $L$ with $\norm{f(t, x_1) - f(t, x_2)} \leq L\norm{x_1 - x_2}$ for $(t, x_1), (t, x_2) \in V$. If $f$ is $\mathcal C^1$ (or merely $\partial_xf$ exists and is continuous), it is locally Lipschitz in $x$: on a compact convex neighborhood, the mean value inequality with $L = \sup\vertiii{\partial_xf}$.