# Proof of continuity: Spherical mean function

1. Mar 11, 2012

### Mugged

This is one of my homework problems.

If h(x) is continuous in x, show that the spherical mean:

M$_{h}$(x,r) = $\frac{1}{w_{n}}$$\int$$_{|\xi|=1}$ h(x+r$\xi$) dS$_{\xi}$

is continuous for all x and r $\geq$ 0.

A lot of PDE textbooks state this fact (in regards to the wave equation in 3 dimensions) - like Evans, McOwen, etc. But none ive looked at provide a proof of continuity.

This should be simply enough but my analysis skills lack. The strategy one would need to take is first take r to be constant, prove continuity in x. Then fix x, and prove continuity in r.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution