When does the limit of the mean values of a function around a point approach the value of the function at that point ?

Question

When does the limit of the mean values of a function around a point approach the value of the function at that point ? We can prove it if the function is continuous. But are there general classes of functions for which this holds ?

In precise language , what is the most general class of functions for which the following hold ?

$\frac{1}{n\alpha(n)\epsilon^{n-1}}\int_{\partial B(x,\epsilon)}f(y)dS(y) \rightarrow f(x)$ as $\epsilon \rightarrow 0 $

Answer 1

The answer to this is the content of the Lebesgue differentiation theorem.