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].
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