Locally Lipschitz Function on an Open Subset of a Metric Space

In the setting of The Real Numbers: Standing Notation and Background, let $(X,d)$ be a metric space, let $\Omega\subseteq X$ be open in $(X,d)$, for $x\in X$ and real $r>0$ let $B_{d}(x,r)$ be the open ball with center $x$ and radius $r$, and let $\psi:\Omega\to\mathbb{R}$.

¶The function $\psi$ is locally Lipschitz on $\Omega$ if for every $x\in\Omega$ there are a positive real number $r$ with $B_{d}(x,r)\subseteq\Omega$ and a nonnegative real number $L$ such that