Exercise 45Calculus › Single-variable calculus › Integration › Riemann integralHigh schoolAdded 5 Nov 2026

Additivity over adjacent intervals

#additivity#step function#split

Statement

Let $f:[a,c]\to\mathbb{R}$ be a step function, and $b\in]a,c[$ . Show that
$\int_a^cf=\int_a^bf+\int_b^cf.$

Hint

Add $b$ to the subdivision of $f$ and compare sums.

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Solution

Refinement. Let $\sigma$ be a subdivision of $[a,c]$ adapted to $f$ . Set $\sigma'=\sigma\cup\{b\}$ : still adapted to $f$ . The sum over $\sigma'$ splits into the sum over the parts of $[a,b]$ and over the parts of $[b,c]$ .

Sum. $\sum_{k}\Delta_kc_k=\sum_{\Delta_k\subseteq[a,b]}c_k\Delta_k+\sum_{\Delta_k\subseteq[b,c]}c_k\Delta_k=\int_a^bf+\int_b^cf$ .

Conclusion. Chasles' relation holds for step functions.

Additivity over adjacent intervals

Statement

Solution

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