Proof that a continuous function is zero

Question

Let $f:[0, 1] \to \mathbb{R}$ be a continuous function such that $\int^1_0 f(x)x^ndx=0$ $\forall n \in \mathbb{N}$. Show that $f \equiv 0$ in [0, 1].

I need help.

Answer 1

Hint

• Using the linearity of the integral and with the hypothesis we get

$$\int_0^1 f(x)p(x)dx=0$$ for every polynomial $p$

• Using the Stone–Weierstrass theorem we get $$\int_0^1f^2(x)dx=0$$ and we conclude the desired result.