# Math Help - intergration

1. ## intergration

suppose f:[a,b]-> real numbers is continuos and for all continuous functions g:[a,b] -> real numbers, then intergrate from b to a of (f.g)=0. show that f is identically 0.

how do i go about solving this?

2. Does f.g means f multiplied by g?

You can pick any g, so let g = f
Now what does the statement simplify to?

Just as a remark, this is true more generally in the sense that if $\displaystyle \int_a^b f(x)h(x)\text{ }dx=0$ for every $h \in C^k[a,b]$ (for some arbitrary but fixed $k\geqslant 0$) with $h(a)=h(b)=0$ then $\displaystyle f\equiv 0$. The proof is not much different, merely take $h(x)=f(x)(x-a)(x-b)$ to conclude. I mention this because it shows that $C^k[a,b]$ separates points with respect to the usual weak topology on it. In this context this theorem (as simple as it is) actually has a name. It's the Fundamental Lemma of the Calculus of Variations.