I believe it is required that be Lebesgue integrable.
See the proof here.
The Lebesgue Differentiation Theorem states:
If is a summable function then for almost every the following limit converges to :
Where is the lebesgue measure and is an open ball of radius .
How does it follow that this Theorem is satisfied if is only assumed to be continuous?
Let me know if anything is unclear. Thanks.