Discontinuous point of a distribution function on ${\Bbb R}^n$

Question

Can the points of discontinuity of a distribution function on $\mathbb R^n$, ($n\gt1$) be uncountable？ What about $\left(\{-\infty\}\cup\Bbb R\cup\{+\infty\}\right)^n$?

Answer 1

I'm not sure what your second sentence means, but here's a hint on your first question: a point of discontinuity of a distribution function has a jump size $\delta > 0$. Create a sequence $\{\epsilon_{_k}\} \searrow 0$. Consider how many points of discontinuity there can be that have jump size between $\varepsilon_{_k}$ and $\varepsilon_{_{k+1}}$ for each $k$, and then consider summing up this number as $k \rightarrow \infty$. How big can this sum be?