Proof of f(x)=0 for All x in [a,b] via Integral Form

[Misswfish]

Suppose f is continuous function on [a,b] such that for each continuous function g, $$\int$$(fg)dj = 0 (Note: integral is from a to b) , then f(x) = 0 for each x in [a,b].

I know that I should use the theorem If is continuous on [a,b], f(x)$$\geq$$0 for each x in [a,b] and theree is a number p i n [a,b] such that f(p) > 0, THen $$\int$$f dj > 0.

I just don't understand how they tie together.

 

[slider142]

I know that I should use the theorem If is continuous on [a,b], f(x)$$\geq$$0 for each x in [a,b] and theree is a number p i n [a,b] such that f(p) > 0, THen $$\int$$f dj > 0.

What is the contrapositive of this statement? It pretty much falls out of it.

 

[Misswfish]

Ahhh my teacher told me to pick a "clever" g(x) so that we can use this theorem and therefore f(x) = 0. I was thinking contradiction but my teacher shot that down