Prove that if f is an increasing real-valued function on an open interval (a, b), then, for all but at most countably many points c in (a, b), Lim_(x-->c) f(x) exists and is equal to f(c).

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The function f is defined and finite at every point of the interval (a, b). Moreover, since every point of an open interval is an interior point, we know that both sup({f(x): x < c, x in (a, b)}) and inf({f(x): x > c, x in (a, b)}) exist and are finite for every c in (a, b), where sup and inf denote supremum and infimum, respectively.

For every c in (a, b), let u_c and l_c denote sup({f(x): x < c, x in (a, b)}) and inf({f(x): x > c, x in (a, b)}), respectively. Since f is increasing, we know that u_c <= l_c for every c in (a, b), and that u_c = l_c if and only if f is continuous at x = c.

If f is continuous ...

Solution Summary

A detailed proof of the following is provided: If f is an increasing real-valued function which is defined on an open interval (a, b), then there are at most countably many points c in (a, b) at which f(x) does not converge to f(c).

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