Exercise 66Calculus › Single-variable calculus › Integration › Riemann integralL1Added 5 Nov 2026

Squeezing between step functions

#Riemann integrability#step functions#squeeze

Statement

Let $f:[a,b]\to\mathbb{R}$ be bounded. Suppose that for every $\varepsilon>0$ there exist step functions $\varphi\le f\le\psi$ with $\int_a^b(\psi-\varphi)<\varepsilon$ . Show that $f$ is Riemann integrable.

Hint

The lower and upper Darboux integrals of $f$ differ by at most $\int(\psi-\varphi)$ .

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Solution

Darboux integrals. For every $\varepsilon>0$ , choose $\varphi\le f\le\psi$ with $\int(\psi-\varphi)<\varepsilon$ . Monotonicity gives
$\int_a^b\varphi\le\int_a^{\underline b}f\le\int_a^{\overline b}f\le\int_a^b\psi,$
so
$0\le\int_a^{\overline b}f-\int_a^{\underline b}f<\varepsilon.$

Conclusion. Since this holds for every $\varepsilon>0$ , the lower and upper Darboux integrals coincide: $f$ is Riemann integrable.

Squeezing between step functions

Statement

Solution

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