Is continuous increasing function in $H^1([0,1])$ Consider a function $f(x):[0,1]\rightarrow \mathbb{R}$. If $f(x)$ is continuous and increasing, is $f(x)$ in $H^1(\Omega)$, the Sobolev space with norm $\sqrt{\int_0^1 (|f(x)|^2 + |Df(x)|^2) dx}$?

As in the comments, the Devil's staircase, $\mathcal D=\mathcal D(x)$ is a non-decreasing function with no weak derivative; therefore it cannot lie in any Sobolev space. Since the OP wants a function that is increasing, one can consider $f(x) := x + \mathcal D(x)$ instead. 
