Lipschitz constant

2020 Mathematics Subject Classification: Primary: 54E40 [MSN][ZBL]

of a function $f$

Definition

For functions $f:[a,b]\to \mathbb R$ it denotes the smallest constant $M>0$ in the Lipschitz condition, namely the nonnegative number \begin{equation*} \sup_{x\neq y} \frac{|f(y)-f(x)|}{|y-x|}\, . \end{equation*} The definition can be readily extended to vector-valued functions on subsets of any Euclidean space and, more in general, to mappings between metric spaces (cf. Lipschitz function).

Relations with differentiability

If the domain of $f$ is an interval, the function is everywhere differentiable and the derivative is bounded, then it is easy to see that the Lipschitz constant of $f$ equals \[ \sup_x |f'(x)|\, . \] A corresponding statement holds for vector-valued maps on convex euclidean domains: in this case $|f'(x)|$ must be replaced by $\|Df (x)\|_o$, where $Df$ denotes the Jacobian matrix of the differential of $f$ and $\|\cdot\|_o$ the operator norm.

A partial converse of this statement is given by Rademacher theorem: a Lipschitz function $f$ on an open subset of a Euclidean space is almost everywhere differentiable and the Lipschitz constant bounds from above the number $S = \sup_x \|Df (x)\|_o$, where the supremum is taken over the points $x$ of differentiability. If in addition the domain is convex, we again conclude that the Lipschitz constant equals $S$.

Remark on terminology

The one used here is the most common terminology. However, some authors use the name Lipschitz condition of exponent $\alpha$ for the Hölder condition \begin{equation}\label{eq:1} |f(y)-f(x)| \leq M |y-x|^{\alpha}\, \end{equation} (cp. with Lipschitz condition). Consistently, such authors use the term Lipschitz constant for the smallest $M$ satisfying \eqref{eq:1} even when $\alpha<1$. For functions satisfying \eqref{eq:1} the most common terminology is instead:

• Hölder exponent for the fixed exponent $\alpha$
• Hölder constant for the smallest $M$ satisfying \eqref{eq:1}.

A common notation for the latter is $[f]_\alpha$.