I want to understand a bit about the weak derivative of singular function Let u be singular. Using wiki to define the concept of singular functions I denote by $N⊂R$ a set such that $u$ has a classical derivative $u'=0$ on $N^c$ and $N$ has measure zero. To compute the weak derivative let $ψ$ be a test function: $$(u',ψ)=-(u,ψ' )=-∫_R uψ' dx=-∫_{N^c} uψ' dx-∫_N uψ' dx$$ **Assume that $N^c$ is just made of decreasingly small intervals so i can integrate by parts again to get: $$(u',ψ)=∫_{N^c} u'ψdx-∫_N uψ' dx$$ Now by definition $u'=0$ on $N^c$ an thus the first integral is zero. The second Is zero to since we integrate a continues function over a set of measure zero. Thus in the weak sense: $$(u',ψ)=0⇒u'=0$$ I already have read this but I just want clarification. Clearly I am doing something wrong since $$(T',ψ)=0⇒T=c$$ Q1. I suspect that the mistake may be at ** but I have no idea if I’m right or not. The best I can come up with is: $N^c$ is obtained by some limiting process which renders integration by parts wrong.
Q2. what is the actual derivative of a Cantor function in the weak sense ? Is it like a Dirac-delta ?