(a) Let $q_{1}, q_{2}, \ldots$ be a fixed enumeration of the rationals in $[0,1]$. For positive reals $a_{1}, a_{2}, \ldots$, define a function $f$ from $[0,1]$ to $\mathbb{R}$ by setting $f\left(q_{n}\right)=a_{n}$ for each $n$ and $f(x)=0$ for $x$ irrational. Prove that if $a_{n} \rightarrow 0$ then $f$ is Riemann integrable. If $a_{n} \rightarrow 0$, can $f$ be Riemann integrable? Justify your answer.

(b) State and prove the Fundamental Theorem of Calculus.

Let $f$ be a differentiable function from $\mathbb{R}$ to $\mathbb{R}$, and set $g(x)=f^{\prime}(x)$ for $0 \leqslant x \leqslant 1$. Must $g$ be Riemann integrable on $[0,1]$ ?