Let $\mu$ be finite Borel measure on $\mathbb{R}^{2}$. For fixed $r>0$, let $C_x=\{y :|y−x|=r\}$ and define $f:\mathbb{R}^{2}\rightarrow \mathbb{R}$ by $f(x)=\mu[C_x]$. Prove that $f$ is continuous at $x_{0}$ if and only if $\mu[C_{x_{0}}]=0$.
I have been trying this exercise but i just couldn't find the relation between the borel measure and the continuity of $f$.