Here is a problem I cannot get my head around. Let $E⊆\mathbb{R}$ be Lebesgue measurable with $\lambda(E)>0$. Consider the distance function $d(x)=\inf\{|t-x|:t\in E\}$. Prove $\lim_{x\rightarrow y}\frac{d(x)}{|x-y|}=0,$ a.e. $y\in E$.
My thoughts... For $x\in E$ we have $d(x)=0$ so we get the limit to be zero. I can improve this for $x\in \bar{E}$, since then we can choose $(x_n)\subseteq E$ with $|x_n-x|\rightarrow 0$, and so $d(x_n)\rightarrow d(x)$ and the limit is zero. But I cannot see what I can do with the general case... Any help?
Thanks in advance...