Exercise 64Calculus › Single-variable calculus › Integration › Riemann integralL1Added 5 Nov 2026

Darboux sums for a step function

#Darboux sums#upper lower#step functions

Statement

Let $f$ be a step function on $[a,b]$ . Show that its lower and upper Darboux sums (over the subdivision adapted to $f$ ) coincide, and deduce that $f$ is Riemann integrable with $\int_a^bf$ defined as usual.

Hint

On the intervals of the adapted subdivision, $\inf f=\sup f=c_k$ .

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Solution

Darboux sums. Take a subdivision $\sigma$ adapted to $f$ . On each $]x_{k-1},x_k[$ , $f=c_k$ , so $\inf=c_k=\sup$ . Hence
$L(f,\sigma)=\sum_kc_k\Delta_k=U(f,\sigma).$

Riemann integrability. Since $\inf_\sigma L(f,\sigma)=\sup_\sigma U(f,\sigma)$ , $f$ is Riemann integrable with value $\int_a^bf$ .

Conclusion. Step functions are Riemann integrable with the expected integral.

Darboux sums for a step function

Statement

Solution

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