Q: If function $f\in C^{1,\alpha} (D)$ (Holder spaces) with D is an open in $\mathbb{R}^2$, then can we have $f$ is piecewise continuos function? If  function $f\in C^{1,\alpha} (D)$ (Holder spaces) with D is an open in $\mathbb{R}^2$, then can we have that $f$ is piecewise continuos function?

A: Every continuous function is  piecewise continuous also ;)
As an example of a function $f\in C^{1,1}$ with $f\not\in C^2$ serves
$$
f(x) = |x|\cdot x
$$
with $f'(x) = |x|$.
