The local Lipschitz continuity condition is standard in **nonsmooth, nonconvex optimization** -- see e.g. [this paper][1] and Sections "Nonsmooth, Nonconvex Optimization" and "The Clarke Subdifferential" in these [lecture notes][2]. 
The Lipschitz continuity condition is also used in **stochastic convex optimization** problems -- see e.g. [this paper][3]. 

The Lipschitz continuity condition plays important roles in [isoperimetric inequalities][4] and [optimal mass transportation theory][5].  


  [1]: https://arxiv.org/pdf/1807.07554
  [2]: https://cs.nyu.edu/overton/conv_ns_opt/notes/CourseLastLecture2018.pdf
  [3]: https://home.ttic.edu/~karthik/nonlinearTR.pdf
  [4]: https://www.google.com/search?q=isoperimetric%20lipschitz&oq=isoperimetric%20lipschitz&aqs=chrome..69i57.9712j0j1&sourceid=chrome&ie=UTF-8
  [5]: https://www.google.com/search?q=optimal%20mass%20transportation%20lipshitz&sxsrf=AOaemvI0VGnT1ioBo63tRyt1Q8rhGgyWNA%3A1640276374382&ei=lqHEYYfjFoOgptQPy6C9yAk&ved=0ahUKEwiHhoPVqfr0AhUDkIkEHUtQD5kQ4dUDCA4&uact=5&oq=optimal%20mass%20transportation%20lipshitz&gs_lcp=Cgdnd3Mtd2l6EAMyBQgAEM0COgcIABBHELADOggIABCwAxCGA0oFCDwSATFKBAhBGABKBAhGGABQrQtY0xlg6yJoAXACeACAAYABiAHOB5IBAzAuOJgBAKABAcgBBcABAQ&sclient=gws-wiz