$f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ $f:[a,b] \to R$ is continuous and $\int_a^b{f(x)g(x)dx}=0$ for every continuous function $g:[a,b]\to R$ with $g(a)=g(b)=0$. Must $f$ vanish identically?

Using integration by parts I got the form:
$\int_a^bg(x)f(x)-g'(x)F(x)=0$. Where $F'(x)=f(x)$.

In particular, putting $g(x)=(x-a)(b-x)f(x)$, we have $\int_a^b{(x-a)(b-x)f(x)^2dx}=0$. The integrand is non-negative in $[a,b]$, so $f = 0$ almost everywhere. As $f$ is continuous, $f$ must be identically zero.
