This question’s been bugging me for over a week now and I’m afraid it’s beyond my capabilities. It concerns the Riemann integral and conditions for it (in a section just after studying the Riemann-Lebesgue Theorem).

Here it is, as stated (‘integrable’ = ‘Riemann integrable’):

Let *f* be integrable on [a,b]. If *g* is bounded and *f* = *g* except for countably many points, must it be true that *g* is integrable on [a,b]?

The answer given is ‘no’, without further explanation. But I can’t see why … it seems *g* must have only countably many discontinuities where *f* = *g*, and potentially countably many where *f* != *g*, which is still only countably many. Then g should be integrable. Unless I’m missing something.

The other thing I thought is that it might be an elaborate “Simon Says” question, where an assumption was made somewhere (e.g. “For the next two chapters, all circles will be regarded as types of squares”) that isn’t supposed to be assumed here. Though I can’t figure out what it would be.