Lipschitz and constant

Lipschitz Function

A function $f$ is called Lipschitz continous (or just Lipschitz) if there exists a constant $L \geq 0$ such that for all points $x$ and $y$ in the domain

\|f(x) - f(y)\| \leq L \cdot \|x - y\|

(or more generally, $\|\|f(x) - f(y)\|\| \leq L \cdot \|\|x - y\|\|$ in higher dimensions).

The output of the function can't change faster than a fixed multiple of how much the input changes. In other words, the function is not allowed to have infinitely steep slopes.

Lipschitz Constant

The number $L$ in the inequality above is called the Lipschitz constant of the function.

The smallest possible $L$ that works for the whole domain is called the best Lipschitz constant or optimal Lipschitz constant.
If $L$ is small, the function changes more slowly/smoothly.
if $L$ is big, the function can change more quickly.

Examples

Function	Is it Lipschitz?	Lipschitz Constant (example)
$f(x) = 5x + 2$	Yes	$L = 5$
$f(x) = \sin(x)$	Yes	$L = 1$ (because the derivative ≤ 1)
$f(x) = x^2$ on $[-1, 1]$	Yes	$L = 2$ (on this interval)
$f(x) = x^2$ on all real numbers	No	Not bounded (gets steeper as \|x\| grows)
$f(x) = \sqrt{x}$ on [0, ∞)	No	Slope becomes infinite near 0