How to show that a bounded, monotone increasing function may only have countably many discontinuities?

Question

Now I've figured it out: if the distance between the upper and lower bound is D then the number of jumps greater than D/n must be less than n, hence finite, and the set of all jumps is the union for all natural n of the sets of jumps greater than D/n, and the union of countably many finite sets is countable. Now my question is: how to show that such a function has limits from the left and right everywhere?

Answer 1

Hmmm.

I remember doing this exercise. The basic idea was that each discontinuity represents a jump: lim(x-)+epsilon=lim(x+) {left and right handed limits}. So how can you have uncountably many jumps, it would have to race off to infinity (i.e. not bounded). But if it had countably many discontinuities then you could have jumps that formed a convergent sequence (like sum (1/n^2)). I hope this helps you get started.

Answer 2

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