For measurable functions $f(x)$, $g(x)$ on $[0,1]$ define the distance $\rho(f,g)$ as a Lebesgue measure of the set $\{x:f(x)\ne g(x)\}$. Then Luzin's famous theorem states that $C[0,1]$ is dense with respect to this metric in the set of all measurable functions.

The question is to describe the completion of $C^1[0,1]$ (that is, informally, for given function give a receipt how to recognize wether it belongs to completion or not.)