Is a C^1function in a bounded domain $\Omega\subset R^n;$ an element of the Sobolev space $W^{2,\infty}(\Omega)$ ?
No. Consider $x^{3/2}$ on $(1,1)$. It has Holder continuous but not Lipschitz derivative. See GilbargTrudinger Ch. 4 for some related exercises (e.g. a C^1 function with derivative continuous but not Holder continuous for any $\alpha$). 

