(a) What does it mean for a function $f:[a, b] \rightarrow \mathbb{R}$ to be Riemann integrable?

(b) Let $f:[0,1] \rightarrow \mathbb{R}$ be a bounded function. Suppose that for every $\delta>0$ there is a sequence

$$0 \leqslant a_{1}<b_{1} \leqslant a_{2}<b_{2} \leqslant \ldots \leqslant a_{n}<b_{n} \leqslant 1$$

such that for each $i$ the function $f$ is Riemann integrable on the closed interval $\left[a_{i}, b_{i}\right]$, and such that $\sum_{i=1}^{n}\left(b_{i}-a_{i}\right) \geqslant 1-\delta$. Prove that $f$ is Riemann integrable on $[0,1]$.

(c) Let $f:[0,1] \rightarrow \mathbb{R}$ be defined as follows. We set $f(x)=1$ if $x$ has an infinite decimal expansion that consists of 2 s and $7 \mathrm{~s}$ only, and otherwise we set $f(x)=0$. Prove that $f$ is Riemann integrable and determine $\int_{0}^{1} f(x) \mathrm{d} x$.