34.12 Show that if f is a continuous real-valued function on [a,b] satisfying
∫ f(x)g(x) dx = 0 for every continuous function g on [a,b], then f(x) = 0 for
all x in [a,b].
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 ...
This is a proof regarding a continuous real-valued function.