Questioning the validity of a proof in Gelfand & Fomin – math.stackexchange.com

I was reading Gelfand&Fomin Calculus of Variations book, in which I found this

Lemma 2 If $\alpha(x)$ is continuous in $[a, b]$, and if $\int_a^b \alpha(x) h'(x)\, dx = 0$ for every function ...