Lipschitz function 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

(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