Discussion
Which of the following statements is not correct ? The set of points of discontinuities of a distribution function may be :
Answer: C
The set of discontinuities cannot be uncountable.
A distribution function is non-decreasing and bounded between 0 and 1. At each discontinuity it jumps by a positive amount, and at most \(k\) jumps can exceed \(1/k\), since the total of all jumps is at most 1. So the jumps can be listed by size, and the set of discontinuities is countable.
All the other possibilities occur: a continuous distribution has none (empty), a binomial has finitely many, and a distribution placing mass \(2^{-n}\) at each \(n\), or at each rational, has countably infinitely many.
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