(D1) A function $f$ is called Lipschitz continuous on a set $A$ if there exists $K \ge 0$ such that, for $x, y \in A$,
$|f(x_1) - f(x_2)| \le K |x_1 - x_2|$
(D2) A function is called locally Lipschitz continuous on $A$ if for each $x \in A$, there is a neighborhood $U_x$ such that $f$ is Lipschitz on $U_x \cap A$.
For my application, I need a condition in between the two:
(D3?) A function is uniformy locally Lipschitz continuous on $A$ if there is a $K \ge 0$ and a $\delta >0$ such that for $x, y \in A$, $|x_1 - x_2| < \delta$ implies
$|f(x_1) - f(x_2)| \le K |x_1 - x_2|$
This definition is especially useful for non-convex sets, in which case we may have $D3$ true but $D1$ untrue.
Is there anything of the sort in the literature?