Suppose that $\mu$ is a signed finite Borel probability measure which is singular with respect to the Lebesgue measure in $[0,1]$. Is it true that there always exist a point $x\in [0,1]$ such that 
 \begin{equation*}
 \lim_{\delta \to 0^+} \frac{\mu([x-\delta,x+\delta])}{\delta } = + \infty \quad \text{or} \quad -\infty \quad ?
\end{equation*}