Consider the functions \(u(x)=\mathbb{1}_{\mathbb{Q} \cap[0,1]}\) and \(v(x)=\mathbb{1}_{\left\{n^{-1}: n \in \mathbb{N}ight\}}(x)\). Prove or disprove the following statements.

(i) The function \(u\) is 1 on the rationals and 0 otherwise. Thus \(u\) is continuous everywhere except for the set \(\mathbb{Q} \cap[0,1]\). Since this is a null set, \(u\) is a.e. continuous, hence Riemann integrable by Theorem 12.9 .

(ii) The function \(v\) is 0 everywhere but for the values \(x=1 / n, n \in \mathbb{N}\). Thus \(v\) is continuous everywhere except for a countable set, i.e. a null set, and \(v\) is a.e. continuous, and hence Riemann integrable by Theorem 12.9 .

(iii) The functions \(u\) and \(v\) are Lebesgue integrable and \(\int u d \lambda=\int v d \lambda=0\).

(iv) The function \(u\) is not Riemann integrable.