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Prove a property of the integral of the product of continuous functions

Let f be a continuous function on the interval [a,b] and assume

    \[ \int_a^b f(x) g(x) \, dx = 0 \]

for every function g which is continuous on the interval [a,b]. Prove that f(x) = 0 for all x \in [a,b].


Proof. Since \int_a^b f(x) g(x) \, dx = 0 must hold for every function g that is continuous on [a,b], it must hold for f itself (since f is continuous on [a,b] by hypothesis). Therefore we must have,

    \[ \int_a^b f(x) f(x) \, dx = \int_a^b (f(x))^2 \, dx = 0 \]

However, (f(x))^2 \geq 0 for all x \in [a,b]. We know from the previous exercise (Section 3.20, #7) that a non-negative function whose integral is zero on an interval must be zero at every point at which it is continuous. By hypothesis f is continuous at every point of [a,b]; hence, f^2 is also continuous at every point of [a,b] (since the product of continuous functions is continuous). Therefore,

    \[ (f(x))^2 = 0 \qquad \text{for all } x \in [a,b]. \]

Which implies,

    \[ f(x) = 0 \qquad \text{for all } x \in [a,b]. \qquad \blacksquare\]

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