A cumulative distribution function (cdf) has a countable set of discontinuity points. They need not be isolated. Let us call non-isolated points 'accumulation points' of this set of discontinuity points. Is it possible that a cdf admits also countably infinite accumulation points in its set of discontinuity points? If so, is this something as pathological as requiring a singular distribution or it can also happen with discrete distributions?
Thanks for any clarification, it would be nice to have concrete examples.