Let
f be a bounded integrable non-negative function continous on [a, b].
Prove:
integral from a to b f(x) dx = 0 implies f(x) = 0 for every x
I believe I understand why it is true, I'm just not sure how to write the formal proof.
Let
f be a bounded integrable non-negative function continous on [a, b].
Prove:
integral from a to b f(x) dx = 0 implies f(x) = 0 for every x
I believe I understand why it is true, I'm just not sure how to write the formal proof.
What have you tried? Here is a weird approach.
Suppose that then there exists some such and since this set is open there exists some open ball and by continuity there exists some such that . In other words, there exists some open ball around the point where which is entirely positive. Take the concentric closed ball . This is a closed and bounded interval and thus, by the Heine-Borel theorem, compact. Thus, since is continuous we know that assumes a minimum on . In particular, . Thus, . This of course is a contradiction.