Let $f : X \to Y$ be a function between metric spaces and $x_0 \in X$. Does the following property have a canonical name? Or is it equivalent to something which is more well-known?
There is a neighborhood $U$ of $x_0$ and a constant $L > 0$ such that $d(f(x_0),f(x)) \leq L \cdot d(x_0,x)$ for all $x \in U$.
I would call this "locally Lipschitz at $x_0$" for the moment. We have the following properties:
- locally Lipschitz at $x_0$ $\implies$ continuous at $x_0$
- Locally Lipschitz $\implies$ Lipschitz on a neighborhood of $x_0$ $\implies$ locally Lipschitz at $x_0$
The converse in the last implication does not seem to hold, since Lipschitz on a neighborhood $U$ of $x_0$ would mean that $d(f(x),f(x')) \leq L \cdot d(x,x')$ holds for all $x,x' \in U$ (not just for $x=x_0$).