Punctual Hölder Continuous

A real or complex valued function \(f\) defined on \(U \subset \mathbb{R}^d\) is called punctual \(\alpha\)-Hölder continuous with \(\alpha>0\), when for every \(x_0\in U\) there are real constants \(C>0\) and \(r>0\), such that for all \(y\in B_r(x_0)\cap U\)

\begin{equation*} |f(x_0)-f(y)|\le C \lVert x-y\rVert^\alpha. \end{equation*}