Basic Analysis I — Real Analysis · Problem 5.1.5

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-integrability

Problem

Suppose

f : [-1,1] \to \mathbb{R}

is defined as

f(x) := \begin{cases} 1 & \text{if } x > 0, \\ 0 & \text{if } x \leq 0. \end{cases}

Prove that

f \in \mathcal{R}([-1,1])

and compute

\int_{-1}^1 f

using the definition of the integral (but feel free to use the propositions of this section).