Lipschitz continuous function

Definition
It is a function $f : (X, d_{X}) \to (Y, d_{Y})$ between two metric spaces which satisfies

d_{Y} (f (a), f (b)) \leq K \cdot d_{X} (a, b)

being $K > 0$ constant.

Remark
Not every continuous function is Lipschitz continuous. For example $f (x) = \sqrt{x}$ .

Remark
This is the condition required for the existence and uniqueness of solutions to ordinary differential equations. See Picard--Lindelöf theorem. Also system of first order ODEs.