Measure, Integration & Real Analysis · Problem 1A.10

Measures — Sheldon Axler

Mathematics — Measure Theory · riemann-integrable-splitting-interval

Problem

Suppose

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

is a bounded function and

c \in (a, b)

. Prove that

f

is Riemann integrable on

[a, b]

if and only if

f

is Riemann integrable on

[a, c]

and

f

is Riemann integrable on

[c, b]

. Furthermore, prove that if these conditions hold, then

\int_a^b f = \int_a^c f + \int_c^b f.