Lipschitz condition

A restriction on the behaviour of increase of a function. If for any points $x$ and $x'$ belonging to an interval $[a,b]$ the increase of a function $f$ satisfies the inequality \begin{equation}\label{eq:1} |f(x)-f(x')| \leq M|x-x'|^{\alpha}, \end{equation}

where $0<\alpha\leq1$ and $M$ is a constant, then one says that $f$ satisfies a Lipschitz condition of order $\alpha$ on $[a,b]$ and writes $f\in\operatorname{Lip}\alpha$, $f\in\operatorname{Lip}_M\alpha$ or $f\in H^{\alpha}(M)$. Every function that satisfies a Lipschitz condition with some $\alpha>0$ on $[a,b]$ is uniformly continuous on $[a,b]$, and functions that satisfy a Lipschitz condition of order $\alpha=1$ are absolutely continuous (cf. Absolute continuity; Uniform continuity). A function that has a bounded derivative on $[a,b]$ satisfies a Lipschitz condition on $[a,b]$ with any $\alpha\leq 1$.

The Lipschitz condition \eqref{eq:1} is equivalent to the condition \begin{equation} \omega(\delta,f)\leq M\delta^{\alpha}, \end{equation}

where $\omega(\delta,f)$ is the modulus of continuity (cf. Continuity, modulus of) of $f$ on $[a,b]$. Lipschitz conditions were first considered by R. Lipschitz [1] as a sufficient condition for the convergence of the Fourier series of $f$. In the case $0<\alpha<1$ the condition \eqref{eq:1} is also called a Hölder condition of order $\alpha$.

References

[1]	R. Lipschitz, "De explicatione per series trigonometricas insttuenda functionum unius variablis arbitrariarum, et praecipue earum, quae per variablis spatium finitum valorum maximorum et minimorum numerum habent infintum disquisitio" J. Reine Angew. Math. , 63 (1864) pp. 296–308
[2]	A. Zygmund, "Trigonometric series" , 1–2 , Cambridge Univ. Press (1988) MR0933759 Zbl 0628.42001
[3]	I.P. Natanson, "Constructive function theory" , 1–3 , F. Ungar (1964–1965) (Translated from Russian) MR1868029 MR0196342 MR0196341 MR0196340 Zbl 1034.01022 Zbl 0178.39701 Zbl 0136.36302 Zbl 0133.31101