Find a $u\in C^1[0,1]$ such that $u\notin H^1(0,1)$, where $H^1(0,1)=W^{1,2}(0,1)$, the Sobolev space.
So finding an example for the case $C^0$ is pretty easy,$u(x)=x^\alpha, 0\lt\alpha\lt\frac{1}{2}$ does the job, but it clearly doesn't work for the other case, since $u'$ is not continuous in 0. Can anyone help me out?