Basic Analysis I — Real Analysis · Problem 5.1.13

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-integral-definition

Problem

Using Exercise 5.1.12 and the idea of the proof in Exercise 5.1.7, show that the Darboux integral is the same as the standard definition of Riemann integral, which you have most likely seen in calculus. That is, show that f∈R([a,b])f \in \mathcal{R}([a,b]) if and only if there exists an I∈RI \in \mathbb{R}, such that for every ϵ>0\epsilon > 0 there exists a δ>0\delta > 0 such that if P={x0,x1,…,xn}P = \{x_0, x_1, \ldots, x_n\} is a partition with Δxi<δ\Delta x_i < \delta for all ii, then ∣∑i=1nf(ci)Δxi−I∣<ϵ\left|\sum_{i=1}^n f(c_i)\Delta x_i - I\right| < \epsilon for every set {c1,c2,…,cn}\{c_1, c_2, \ldots, c_n\} with ci∈[xi−1,xi]c_i \in [x_{i-1}, x_i]. If f∈R([a,b])f \in \mathcal{R}([a,b]), then I=∫abfI = \int_a^b f.

Related theory

Browse the problem library