Basic Analysis I — Real Analysis · Problem 5.1.6

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · riemann-integrability

Problem

Let

c \in (a,b)

and let

d \in \mathbb{R}

. Define

f : [a,b] \to \mathbb{R}

f(x) := \begin{cases} d & \text{if } x = c, \\ 0 & \text{if } x \neq c. \end{cases}

Prove that

f \in \mathcal{R}([a,b])

and compute

\int_a^b f

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