Lipschitz function

Definition

A Lipschitz function is a function $f$ such that

for all $x$ and $y$ (or for all $x$ and $y$ in the set if bound), where $C$ is a constant independent of $x$ and $y$

(Lipschitz continuous)

more general form:

Given two metric spaces $(X,d_X)$ and $(Y,d_Y)$, where $d_X$ denotes the metric on the set $X$ and $d_Y$ is the metric on set $Y$, a function $f: X → Y$ is called Lipschitz continuous if there exists a real constant $K ≥ 0$ such that, for all $x_1$ and $x_2$ in $X$,

G-Lipschitz

for all $\mathbf{x}$, $||\nabla f(\mathbf{x})||_2 \leq G$

(note: norm of gradient, )

Properties:

differentiable at x ⇒ Lipchitz continuous at x ⇒ continuous at x (but the converse is not true)

References:

1. https://www.chrismusco.com/amlds2023/lectures/lec8_annotated.pdf
2. https://mathworld.wolfram.com/LipschitzFunction.html
3. https://math.berkeley.edu/~mgu/MA128ASpring2017/MA128ALectureWeek9.pdf
4. https://en.wikipedia.org/wiki/Lipschitz_continuity
5. https://users.wpi.edu/~walker/MA500/HANDOUTS/LipschitzContinuity.pdf