Assume $f\in\mathcal{C}[a,b]$ and that for all $h\in\mathcal{C}[a,b]$ wich is zero at the endpoints it holds that $$\int f(x) h(x) d x =0$$. Then $f(x)=0$ for all $x \in [a,b]$

Proof

HINT: Proof by contradiction, assuming that $f$ is non zero somewhere.

Remark This is analogous to the vector algebra proposition, let $v,w\in V$, where $V$ is a vector space. If $v\cdot w = 0$ for all $w\in V$ then $v=0$. In fact, if you dig deeper, it is the same result: the space of continuous functions from $a$ to $b$ is a vector space, and we can define the integral of the multiplication as the inner product!

References

• Calculus of Variations - Gelfand and Fomin.