View source for 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|Absolute continuity]]; [[Uniform 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|Continuity, modulus of]]) of $f$ on $[a,b]$. Lipschitz conditions were first considered by R. Lipschitz [[#References|[1]]] as a sufficient condition for the convergence of the [[Fourier series|Fourier series]] of $f$. In the case $0<\alpha<1$ the condition \eqref{eq:1} is also called a [[Hölder condition|Hölder condition]] of order $\alpha$.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  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  {{MR|}} {{ZBL|}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''1–2''' , Cambridge Univ. Press  (1988)  {{MR|0933759}} {{ZBL|0628.42001}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  I.P. Natanson,   "Constructive function theory" , '''1–3''' , F. Ungar  (1964–1965)  (Translated from Russian)  {{MR|1868029}} {{MR|0196342}} {{MR|0196341}} {{MR|0196340}} {{ZBL|1034.01022}} {{ZBL|0178.39701}} {{ZBL|0136.36302}} {{ZBL|0133.31101}} </TD></TR></table>