For which $p\in\Bbb{R}$ exists a signed measure $\mu$ on [0,1] with the sigma-algebra of the Borel sets with $\mu([0,x])=f(x):=x^psin(1/x)$. Find a Hahn decomposition for $\mu$.
I find the part with $p$ quite difficult but the Hahn decomposition I consider impossible. If I understand all informations that were given to me correctly then the Hahn decomposition separates intervalls where f(x) is monotonically increasing from those where f(x) is monotonically decreasing. Therefore I would need the extrema. Whereas the solutions of f(x)=0 are easy the solutions of f '(x)=0 can only be approximated numerically which is a problem with an infinite number of solutions.