Hello,

The question might be too naive. I am just not very confident about it. Let $f$ be a singular continuous function on $R$. We have $\int f \psi d x=0$ for any test function $\psi\in\mathcal{D}(R)$ (smooth functions with compact support). So the singular functions are vanishing distributions. Am I right? 

Thank you very much for any hints!


Anand


p.s. this is related to [my previous post][1].


  [1]: http://mathoverflow.net/questions/70146/distributions-and-measures