finding a function $f$ such that the following holds:
$f\in lip_0^{\alpha}[0,1] $ where $\alpha =\frac {1}{2}$ and base point is zero, such that $\tilde{f}=\frac {f(x)-f(y)}{ \vert x-y \vert^{1/2} }$($\tilde {f}$ is in $C_0(\tilde X); \tilde X=\{(x,y) | x\neq y\} $) has the following property:
if $ (x_0, y_0) $ in $\tilde X $ , for any $\epsilon >0 , $open neighborhood U of $x_0$ ,V of $y_0$
$\Vert \tilde{f}\Vert_{sup}=1=\tilde{f}(x_0,y_0)$ ,
$\vert \tilde{f}(x,y) \vert< \epsilon $ ; $(x,y)\in (U^C\cap V^C) $
