Lipschitz class

Remarks

Example: $f(x) = \sqrt{x}$ on $[0, 1]$ lies in $\mathrm{Lip}_1 \tfrac12$. For $0 \le y \le x$ one has $\sqrt{x} - \sqrt{y} \le \sqrt{x - y}$, since squaring gives $x + y - 2\sqrt{xy} \le x - y$, that is $y \le \sqrt{xy}$.

Non-example: the same $f$ is not in $\mathrm{Lip}\,1$, because $\abs{f(x) - f(0)} / \abs{x - 0} = x^{-1/2}$ is unbounded as $x \to 0$.

In terms of the modulus of continuity $\omega(f, \delta)$, the condition $f \in \mathrm{Lip}_M \alpha$ says $\omega(f, \delta) \le M \delta^\alpha$ for all $\delta > 0$. $\mathrm{Lip}\,1$ is the class of Lipschitz continuous functions, and $\mathrm{Lip}\,\alpha$ for $\alpha < 1$ is the class of Hölder continuous functions of exponent $\alpha$. The restriction $\alpha \le 1$ is natural: for $\alpha > 1$ the condition forces $f' = 0$, so only constant functions remain.