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 and $g(u) = 0$ otherwise. Hence $\int_x^{x+y} f(u) du = 0.$ Hence $\frac{\int_x^{x+y} f(u) du}{y} = 0.$ tend $y = 0$ we get $f(x) = 0.$ Hence proved.
Is this correct?