Exercise 63Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

Integral of $|f|$ for a step function

#absolute value#step function#triangle inequality

Statement

Let $f$ be a step function on $[a,b]$ . Show that $|f|$ is a step function and
$\left|\int_a^bf\right|\le\int_a^b|f|.$

Hint

$|f|$ has the same subdivisions as $f$ ; apply the triangle inequality to the sum.

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$|f|$ is a step function. On each interval where $f=c_k$ , $|f|=|c_k|$ , constant.

Conclusion. The triangle inequality for integrals holds for step functions.