(a) State the Weierstrass approximation theorem concerning continuous real functions on the closed interval $[0,1]$.

(b) Let $f:[0,1] \rightarrow \mathbb{R}$ be continuous.

(i) If $\int_{0}^{1} f(x) x^{n} d x=0$ for each $n=0,1,2, \ldots$, prove that $f$ is the zero function.

(ii) If we only assume that $\int_{0}^{1} f(x) x^{2 n} d x=0$ for each $n=0,1,2, \ldots$, prove that it still follows that $f$ is the zero function.

[If you use the Stone-Weierstrass theorem, you must prove it.]

(iii) If we only assume that $\int_{0}^{1} f(x) x^{2 n+1} d x=0$ for each $n=0,1,2, \ldots$, does it still follow that $f$ is the zero function? Justify your answer.