Basic Analysis I — Real Analysis · Problem 5.2.16

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · darboux-integrals

Problem

Suppose

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

and

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

are bounded. a) Show

\overline{\int_a^b}(f+g) \leq \overline{\int_a^b} f + \overline{\int_a^b} g

and

\underline{\int_a^b}(f+g) \geq \underline{\int_a^b} f + \underline{\int_a^b} g

. b) Find example

f

and

g

where the inequality is strict. Hint:

f

and

g

should not be Riemann integrable.