📂MetricSpace

Lipschitz Continuity

Definition¹

For two metric spaces $(X, d_{X})$ and $(Y, d_{Y})$, let’s assume that a function $f : X \to Y$ is given. If there exists a constant $K$ such that the following holds for all $x_{1}, x_{2} \in X$, then $f$ is called $K$-Lipschitz continuous.

$$d_{Y} \big( f(x_{1}), f(x_{2}) \big) \le K d_{X} \big( x_{1}, x_{2} \big)$$

Such a constant $K$ is called the Lipschitz constant.

Explanation

It is named after the German mathematician Rudolf Lipschitz. Saying that $f$ is Lipschitz continuous means that there is a maximum value for the average rate of change. If it is Lipschitz continuous for all open balls, it is called locally Lipschitz continuous. If $X, Y$ is a Euclidean space, then,

$$\left| f(x_{2}) - f(x_{1}) \right| \le K \left| x_{2} - x_{1} \right|$$

It’s a condition related to the stability of solutions in numerical analysis of differential equations.

Properties

If $f : \mathbb{R} \to \mathbb{R}$ is $K$-Lipschitz continuous, then $f$ is:

• Absolutely continuous.
• Differentiable almost everywhere.
• Almost everywhere $\left| f^{\prime} \right| \le K$.

A differentiable $f$ being Lipschitz continuous is equivalent to $f^{\prime}$ being a bounded function.