Absolute value preserves Riemann integrability

Let $f:[a,b]\to\mathbb{R}$ be Riemann integrable .

Proposition: The function $|f|$ is Riemann integrable on $[a,b]$. Moreover, $\left|\int_a^b f(x)\,dx\right|\le \int_a^b |f(x)|\,dx.$

This is a basic stability property: composing with the continuous map $x\mapsto |x|$ preserves Riemann integrability.

Proof sketch: Since $x\mapsto |x|$ is continuous, the discontinuity set of $|f|$ is contained in the discontinuity set of $f$ (continuity points of $f$ remain continuity points of $|f|$). By the Lebesgue criterion , $f$ has measure-zero discontinuity set, hence so does $|f|$, so $|f|$ is integrable. The inequality follows from $-\lvert f\rvert\le f\le \lvert f\rvert$ and monotonicity of the integral.