# Continuous function

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34.12 Show that if f is a continuous real-valued function on [a,b] satisfying

b

∫ f(x)g(x) dx = 0 for every continuous function g on [a,b], then f(x) = 0 for

a

all x in [a,b].

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This is a proof regarding a continuous real-valued function.

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Please see the attachment.

We need to use the following properties of integrals. Suppose is integrable on the interval .

P1: If for any , then

P2: For any , we have

P3: If for any , then .

Suppose is a continuous function on and satisfies for every continuous ...

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