Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant.

Question

Show that if $\int_0^1 f(x) v(x) dx = 0$ for every function v for which $\int_0^1 v(x) dx = 0$, then f is constant. I do not know how to do it.

Answer 1

Hint: Let $v(x)=f(x)-\int_0^1 f(t)\,dt$.