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

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

is continuous for all x and r [itex]\geq[/itex] 0.

A lot of PDE textbooks state this fact (in regards to the wave equation in 3 dimensions) - like Evans, McOwen, etc. But none I've 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.

Please help me