Lebl-Basic-Analysis-I · Problem 5.2.3

The Riemann Integral

Mathematics · riemann-integrability

Problem

Let f:[a,b]→Rf : [a,b] \to \mathbb{R} be Riemann integrable, and g:[a,b]→Rg : [a,b] \to \mathbb{R} be such that f(x)=g(x)f(x) = g(x) for all x∈(a,b)x \in (a,b). Prove that gg is Riemann integrable and that ∫abg=∫abf.\int_a^b g = \int_a^b f.
Intuitive explanation

The key insight is that Riemann integrability depends only on the function's behavior on the interior of the interval—the values at the endpoints are negligible. Since ff and gg agree everywhere except possibly at aa and bb, they have identical upper and lower sums for any partition, making gg integrable with the same integral as ff.

Intuitive explanation

The key insight is that Riemann integrability depends only on the function's behavior on the interior of the interval—the values at the endpoints are negligible. Since ff and gg agree everywhere except possibly at aa and bb, they have identical upper and lower sums for any partition, making gg integrable with the same integral as ff.

Related theory

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