24 1. The Regulated and Riemann Integrals

(3) Suppose f and g are Riemann integrable functions defined

on [a, b]. Prove that if h(x) = max{f(x), g(x)}, then h is

Riemann integrable. This generalizes to the max of a finite

set of functions, but not of infinitely many. Show there exists

a family {fn}n∈N of step functions such that for each n and

each x ∈ [a, b] the value of fn(x) is either 0 or 1 and yet the

function defined by g(x) = max{fn(x)}n∈N is not Riemann

integrable.

(4) Prove that if f and g are bounded Riemann integrable func-

tions on an interval [a, b], then so is fg. In particular, if

r ∈ R, then rf is a bounded Riemann integrable function

on [a, b].