Limit of Average Values of a Continuous Function

A recent homework assignment asked to prove the following fact: If \(f : B \to \mathbf{R}\) is continuous where \(B \subset \mathbf{R}^n\) is open, and \(x_0 \in B\) then
\[
\lim_{r \to 0} \frac{1}{V(B_r)} \int_{B_r} f(x), dV = f(x_0)
\]
where \(B_r\) is the ball of radius \(r\) centered at \(x_0\). The limit-and can be interpreted as the average value of \(f\) on \(B_r\). Informally, the claim says that the average value of a continuous function tends to the value of the function as we consider smaller and smaller balls containing \(x_0\).

There were some problems with the solutions I saw on the homework, so I wrote up a careful solution available here. In general, when trying to prove a fact such as this, a good starting point is to write the definitions of terms in the hypotheses and conclusion. In this case, you need to start with the definitions for continuity, limit, and volume in order to have a fighting chance of proving the statement.

Will Rosenbaum

Tel Aviv

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