Property of a Double Integral involving a limit

[erogard]

Hi,

I am actually not really concerned about what the whole details are but more whether my approach is correct to show the following statement:

Let $$f$$ be continuous on a closed bounded region $$\Omega$$ and let $$(x_0 ,y_0)$$ be a point in the interior of $$\D_r$$. Let $$D_r$$ be the closed disk with center $$(x_0 ,y_0)$$ and radius $$r$$. Then

$$\displaystyle\lim_{r\to 0}\frac{1}{\pi r^2} \displaystyle\int\displaystyle\int_{D_r} f(x,y)dx dy= f(x_0 , y_0)$$

(see $$D_r$$ as the region of the entire double integration)
I have thought of considering a "Riemann sum" approach, which seems a little too brutal and complicated to me (although might do well with the limit involved) or using the MVT for double integrals, which invokes the same hypotheses. Does the latter sound like a good idea?

 

[HallsofIvy]

Yes, the mean value theorem. In fact, the left side, before the limit, is the "mean" (average) value of f(x,y) over the disk.

 

[erogard]

OK, and I almost figured it out. Thank you.