Suppose $f$ is a continuous function on $[a,b]$ and $$ \int_a^b f(x)g(x) = 0$$ for every integrable function. Show that $f(x) = 0$ on $[a,b]. $
Here is what I have so far:
Consider any $x \in [a,b].$ Consider any $y >0.$ Say $g(u) = 1$ for $x
Is this correct?