Modulus of continuity

Summary

A modulus of continuity of a function $f$ is a function $\omega$ that bounds how much $f$ can change: $\abs{f(x) - f(y)} \le \omega(\abs{x - y})$. The best such bound is the function $\delta \mapsto \sup \set{\abs{f(x) - f(y)} : \abs{x - y} \le \delta}$. A function is uniformly continuous exactly when this bound tends to $0$ with $\delta$. Moduli of continuity measure the quality of continuity, and Lipschitz and Hölder conditions are moduli of special forms.