# Continuous function

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].

https://brainmass.com/math/graphs-and-functions/continuous-real-value-function-31264

#### Solution Preview

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 ...

#### Solution Summary

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

$2.19