Let be an integrable, non-negative function on the interval . Prove that if

then at each point at which is continuous.

* Proof. * The proof is by contradiction. Let be a point at which is continuous, and suppose . This implies since is non-negative by hypothesis. Since is continuous at , we know by the sign-preserving property of continuous functions (Theorem 3.7 in Apostol), that there is some neighborhood about , say such that has the same sign as for all , i.e., such that for all . But then by the monotone property of the integral we know,

But then,

This is a contradiction since is nonnegative we know

Since is strictly positive, the sum of the three pieces of the integral cannot equal

The theorem and the proof are good and useful, but I suggest adjusting the title. The note about continuity is important.