Page 284 Problem 6
Let f be continuous on [a, b] and suppose that, for every integrable function g defined on [a, b],
Rb
a f g = 0. Prove that f (x) = 0 for all x ∈ [a, b].
Solution
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The equality a f g = 0 holds for every integrable function g, so letting g = f gives a f 2 = 0. Since
(f (x))2 ≥ 0 for all x ∈ [a, b], the contrapositive of problem 5 gives (f (x))2 = 0 for all x ∈ [a, b],
from which it follows that f (x) = 0 for all x ∈ [a, b].