Proving the continuity of an integral function

Question

I have the following function: $$ J(r,y)=\lvert\mathbb{S}^{n-1}\rvert^{-1} \int_{\mathbb{S}^{n-1}}\lvert r\omega - y \rvert^{2-n} d\omega $$ for $n\ge 3$, where $\mathbb{S}^{n-1}$ is the unit sphere and $\lvert\mathbb{S}^{n-1}\rvert$ is it's measure. I am supposed to show, that this expression is equal to $\min(r^{2-n},\lvert y \rvert^{2-n})$. For $\lvert y \rvert > r $ this can be done by harmonicity of the integrand, and for $\lvert y \rvert < r $ by symmetry. Now all that is left is to prove the continuity of the $J(r,y)$ in $r$ and $y$ to conclude, that this is also true for $\lvert y \rvert = r$. Sadly I am stuck here. It seems to me, that I should pick sequence $(r_n,y_n)_{n \in \mathbb{N}}$ and apply the dominated convergence theorem, but I am unable to construct an integrable, dominating function. If anybody could nudge me in the right direction, it would be much appreciated.