The local Lipschitz continuity condition is standard in nonsmooth, nonconvex optimization -- see e.g. this paper and Sections "Nonsmooth, Nonconvex Optimization" and "The Clarke Subdifferential" in these lecture notes.
The Lipschitz continuity condition is also used in stochastic convex optimization problems -- see e.g. this paper.
The Lipschitz continuity condition plays important roles in stochastic differential equations, isoperimetric inequalities, and optimal mass transportation theory.