>[!abstract] >A Lipschitz condition, named after mathematician Rudolf Lipschitz, is a regularity condition used in mathematical analysis and differential equations that limits the "smoothness" of a function. It essentially requires that the function's change between any two points is bounded by a constant multiple of the distance between those points. This condition is crucial in various fields, including differential equations, functional analysis, and computer science, particularly for ensuring the existence and uniqueness of solutions to certain problems. >[!note] >To satisfy a Lipschitz condition, a function must obey the following: >- Boundedness of slope: the rate of change of the function must remain bounded. >- Existence and uniqueness: in differential equations, solutions to the function must exist and be unique. >- Continuity: the function must be continuous. > > For example, $f(x)=\sqrt(x)$ is not Lipschitz on the interval $[0, \infty]$ because its derivative (slope or rate of change) approaches $\infty$ as $x$ approaches $0$. >[!related] >- **North** (upstream): [[Continuity]] >- **West** (similar): [[Hölder condition]] >- **East** (different): — >- **South** (downstream): —