Examples 7.1.17(a):

Show that every monotone function defined on [a, b] is Riemann integrable.
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We have shown before that a monotone function f defined on a closed interval [a, b] has at most countably many discontinuities.

Therefore such a function f is continuous except at countably many points, so that by our previous theorem the function must be Riemann integrable.

Interactive Real Analysis, ver. 1.9.5
(c) 1994-2007, Bert G. Wachsmuth
Page last modified: Mar 28, 2007