Give examples of the following functions.
(a) $f: [a, b] \to \mathbb{R}$ has countably infinitely many discontinuities.
(b) $f: [a, b] \to \mathbb{R}$ is integrable but neither continuous nor monotone.
(c) $f, g: [a, b] \to \mathbb{R}$ are not integrable, but the sum $f + g$ is integrable.
(d) $f, g: [a, b] \to \mathbb{R}$ are not integrable, but the product $fg$ is integrable.
(e) $f: [c, d] \to \mathbb{R}$ and $u: [a, b] \to [c, d]$ where $u$ is not integrable but $f \circ u$ is integrable.
Let $f: [a, b] \to \mathbb{R}$ be monotone decreasing. Prove $f$ is integrable.
