TITLE: Let $f$ be integrable on $[a,b]$ and suppose for each integrable function $g$ defined on $[a,b]$, $\int^{b}_afg=0$, then $f(x)=0,\forall x\in[a,b]$
QUESTION [6 upvotes]: I do not think this is true,
but at the same time I am not sure.
I know that if we assume that f is continuous instead of integrable then this statement is true. I just do not know how to provide a counterexample if it is false to show that this is wrong. Integrable does not imply continuity I know that much.

REPLY [2 votes]: Okay since this is true for every integrable function g, take $g=f$
Now this becomes $\int^{b}_aff=0$
Now we know that $f^2(x)\geq0$
and $\int^{b}_af^2(x)=0$, This means that the area of the curve is zero even if the function is always above x axis, only one conclusion can be derived from this, that is 
$f^2(x) \equiv 0$ $\forall$ x $\in$ $(a,b)$
The function becomes identically zero.
therefore $f(x)=0,\forall x\in[a,b]$
EDIT: If it is given that f(x) and g(x) is continuous functions then this approach would work, since there is no such condition in this question, hence the statement becomes false.