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.