evil derivative

It is hard to write down a precise analytical description of the $\varphi_k$ here, but the basic form they can take isn't too hard to describe. The problem is to find $\varphi_k$ with the property that $$\int_0^1 \varphi_k = 0 \quad \text{and} \quad \int_0^1 u \, \varphi_k' = -1$$for all $k$, and such that $\varphi_k$ converges to $\chi_{(0,1)}$.

Do the following:

1. let $\varphi_k$ be identically $1$ on $(1/k,1-1/k)$. This takes care of the convergence.
2. on $(1-1/k,1)$ give $\varphi_k'$ a dip to a large negative number and then a bounce back up to $0$
3. define $\varphi_k$ on $(0,1/k)$ so that its integral vanishes on $(0,1)$.

Varying the size of the dip will change the value of the integral of $u \, \varphi_k'$ since this functions is weighted more heavily near $1$ than near $0$. The technical challenge is to make the dip just right so that $u \, \varphi_k'$ has integral exactly $-1$. It isn't hard to see this is possible.

Finally this leads to $$\int_0^1 u \, \varphi_k' = -1$$ for all $k$ giving you a nonvanishing distibutional derivative of $u$.

This question (and @Umberto P.'s good answer) nicely illustrates how distributions/generalized functions explain that the seemingly paradoxical features of the Cantor-Lebesgue staircase function are not paradoxical after all.

Another approach, is via Sobolev spaces and Sobolev imbedding. That is, for a distribution $u$ such that $u\in L^2[0,1]$ and its distributional derivative $u'$ is also in $L^2[0,1]$, we show not only the Sobolev imbedding $u\in C^{0,{1\over 2}}$ (with Lipschitz index ${1\over 2}$), but that, incidentally, that such functions $u$ do satisfy the fundamental theorem of calculus (which the Cantor-Lebesgue function is designed to fail). Thus, for example, the Cantor-Lebesgue function, while in $L^2$, evidently does not have distributional derivative in $L^2$.