Proof of continuity: Spherical mean function

[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 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

 

[lippyka]

prove this. Thanks!Proof: We will prove continuity of M_h(x,r) by showing that the function is both continuous in x and r. First, let r be constant and consider the limit as x approaches x_0. Note that h(x) is continuous in x so that h(x) -> h(x_0) as x -> x_0. Additionally, since |ξ| = 1 for all ξ, the surface integral of h(x+rξ) is also continuous with respect to x. Therefore, as x -> x_0, we haveM_{h}(x,r) -> M_{h}(x_0,r)where M_{h}(x_0,r) is a finite number since h(x_0) is a finite number. Thus, M_{h}(x,r) is continuous in x for fixed r.Next, let x be constant and consider the limit as r -> 0. As r -> 0, the spherical mean can be written asM_{h}(x,r) = \frac{1}{w_{n}}\int_{|\xi|=1} h(x) dS_{\xi} + \frac{1}{w_{n}}\int_{|\xi|=1} (h(x+r\xi)-h(x)) dS_{\xi}Since h(x) is a finite number, the first term on the right hand side goes to a finite number as r -> 0. The second term on the right hand side can be written as\frac{1}{w_{n}}\int_{|\xi|=1} \left(\frac{h(x+r\xi)-h(x)}{r}\right) r dS_{\xi}.Note that the integrand can be written as a product of a continuous function and r. Since the integral is taken over a bounded domain and both terms go to zero as r -> 0, we can apply the Dominated Convergence Theorem to conclude that the integral, and thus the second term, goes to zero as r -> 0. Hence,