Continuity of $m(E\cap B(x,r))$ with respect to $r.$

Question

I was a little curious about continuity of the lebesgue measure. There was a question about showing that for all $E\subseteq \mathbb{R}$ with $0<m(E)<\infty,$ to show that for all $a\in(0,1)$ there exists $I$ an interval with $m(E\cap I)=am(I)$. I was thinking to by Lebesgue Differentiation choosing a point $x$ with $$m(E\cap B(x,r))/m(B(x,r))\to 1.$$ Then as $m(E)$ is finite, we can choosing $n$ large we find $$m(E\cap B(x,r))/m(B(x,r))<a.$$ Now assuming continuity with respect to $r$ the desired result should just be the intermediate value theorem, correct?