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For example consider \(f(x)=x \sin(1/x) 𝟙_{\mathbb{R}\setminus\{0\}}(x)\). This is Lipschitz in \(x=0\) and therefore punctual 1-Hölder continuous in \(x=0\). In every other point this function is differentiable and thus puctual 1-Hölder continuous.

Consider \(x_0=0\). We choose \(n>0\) such that \(C<2n\) and \(x,y0\) and \(r>0\). Then we obtain

\begin{equation*} |f(x)-f(y)| = \frac{2n}{\pi(n^2-\frac{1}{4})}>C \frac{1}{\pi(n^2-\frac{1}{4})}=C|x-y|. \end{equation*}

Thus, \(f\) is not local 1-Hölder continuous.