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Lipschitz continuity

Lipschitz continuity is a stronger form of continuity for a function. A function is Lipschitz continuous if there’s a bound on how fast it can change.

Mathematical Definition

A function $f: D \mapsto R$ is Lipschitz continuous if there exists a constant $L \geq 0$ (called the Lipschitz constant) such that for all $x_1, x_2$ in the domain $D$:

$$ |f(x_1) - f(x_2)| \leq L|x_1 - x_2| $$

Intuitive Understanding

  1. The function’s rate of change is bounded
  2. Geometrically, the absolute slope between any two points doesn’t exceed $L$
  3. The function’s graph cannot have any “vertical” portions

Examples

Basic Functions

  • $f(x) = x$ is Lipschitz continuous with $L = 1$
  • $f(x) = x^2$ is:
    • Lipschitz continuous on any bounded interval
    • Not Lipschitz continuous on $\mathbb{R}$
  • $f(x) = \sqrt{x}$ is not Lipschitz continuous on $[0,\infty)$ due to infinite derivative near 0

Advanced Examples

  • $f(x) = \sin(x)$ is Lipschitz continuous with $L = 1$
  • $f(x) = e^x$ is:
    • Lipschitz continuous on any interval $(-\infty, M]$
    • Not Lipschitz continuous on $\mathbb{R}$

Key Properties

  1. Every Lipschitz continuous function is uniformly continuous
  2. For differentiable functions:
    • $f$ is Lipschitz continuous $\iff$ $f’$ is bounded
    • The smallest valid Lipschitz constant $L$ equals $\sup_{x \in D} |f’(x)|$
  3. On a compact domain, any $\mathcal{C}^1$ function is Lipschitz continuous

Generalizations

  1. Hölder Continuity: A weaker condition where: $$|f(x_1) - f(x_2)| \leq L|x_1 - x_2|^\alpha$$, $\alpha \in (0,1]$.

  2. Vector-valued Functions: For $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$: $$|f(x_1) - f(x_2)| \leq L|x_1 - x_2|.$$