continuity of $J(r,y) = |S^{n-1}|^{-1} \int_{\mathbb{R}^n} |r \omega - y|^{2-n} \text{d} \omega$

Question

In a proof I'm trying to understand the continuity of this function is just being assumed and I'm a little stuck trying to show why. This function is defined on $ \mathbb{R}^{n+1} $ with r the radius and y an element of $ \mathbb{R}^n $

Origin: Analysis by Elliott H. Lieb, Theorem 9.7