Locally Lipschitz map

A map between metric spaces is locally Lipschitz if each domain point has a neighborhood on which the map is Lipschitz. The neighborhood and its constant may depend on the point.

State variables and parameters

For $F(t,y)$ , local Lipschitz continuity in $y$ , locally uniformly in $t$ means that near each $(t_0,y_0)$ , a single finite $L$ satisfies $|F(t,y)-F(t,z)|\le L|y-z|$ for all allowed $t,y,z$ . This is the condition used in the local ODE theorem. A continuous state derivative $D_yF$ supplies it by the mean value estimate on a small convex ball. Separate pointwise constants without local uniform control do not state the same hypothesis.