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,
This is a contradiction since is nonnegative we know
Since is strictly positive, the sum of the three pieces of the integral cannot equal