Is the same thing true for a monotone function defined for allreals? Solution Th

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The hint solves the problem. For each positive integer n,look at the set {x : f(x+) - f(x-) >= 1/n}. This isclearly a finite set, since f is increasing on a finite measureinterval [a,b], so it has upper bound of f(b). Now, the setof discontinuous points is just the union of these sets over n = 1,2, 3, 4, ..., and this is a countable union of finite sets, whichis at most countable. It is also true if f is defined for all reals and ismonotone. For any n, we can restrict f to [-n,n], and thisrestriction (by what we proved above) has at most countablediscontinuities. The set of points where f is discontinuousis just the union of these as n = 1, 2, 3, 4, ..., and this isagain at most countable.

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