I see that Lipschitz continuity is a common assumption used in optimisation, statistics, machine learning, etc. 

Could you point me in the direction of some literature that discusses why Lipschitz continuity is commonly assumed? Does it naturally occur in applications? Contains an important class of functions? 

EDIT: I’ve found that it’s central for ODEs because of Picard-Lindelöf theorem but I’d like to have something closer to statistics or optimization.