[Math] Riemann integrable functions: $\left|\int_a^bf(x)\,\mathrm dx\right|\le\int_a^b|f(x)|\,\mathrm dx$

Question

Suppose that $f\colon [a,b]\rightarrow \mathbb R$ is bounded and has finitely many discontinuities. Show that as a function of $x$ the expression $|f(x)|$ is bounded with finitely many discontinuities and is thus Riemann integrable. Then show that:
$$\left|\int_a^bf(x)\,\mathrm dx\right|\le\int_a^b|f(x)|\,\mathrm dx.$$
I have no idea where to even begin this problem.

Answer 1

hint: $|f(x)| = \max(f(x), 0) - \min(f(x), 0)$

hint: $f(x) = \max(f(x), 0) + \min(f(x), 0)$

hint: if f and g are Riemann integrable functions, then so is $f+g$, $\int (f+g) = \int f + \int g$