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Is it true that a subset $U \subset \mathbb{R}$ is uncountable if and only if it contains an interval? I feel like this should be so but there could be some wonky counterexample.. Thanks!

edit: The motivation for my question comes form "An introduction to Mathematical Statistics and its Applications by Larsen and Marx".

Definition 3.2.1: A real valued function whose domain is the sample space $S$ is called a random varaible.

The author continues:

"If the range of the mapping is either a finite or countably infinite number of values, the random variable is said to be discrete; if the range includes an interval of real numbers, bounded or unbounded, the random variable is said to be continuous"

Asaf Karagila
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    What about the irrationals? – lulu Jul 10 '21 at 23:30
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    If you know any countable set which is dense in $\mathbb{R}$ then take that out and see what's left. – dxiv Jul 10 '21 at 23:31
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    I edited my question. Seems like the author didn't consider if a random variable had it's range as the irrationals when he defined a discrete random variable. –  Jul 10 '21 at 23:34
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    I can't speak specifically to this author, but it's pretty common for textbooks to leave a huge gap between purely atomic probability distributions and probability distributions that have a density, when first introducing this topic. Those categories are by no means exhaustive, but it seems that you may need measure theory to make sense of some of the middle-ground... – Brian Moehring Jul 10 '21 at 23:43
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    I don't believe they were trying to make a general statement about subsets of the reals. Most standard probability distributions have support that falls into one or the other type they mention. That's all they mean. – lulu Jul 11 '21 at 00:25
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    Strictly speaking you can have $X:[0,1]\rightarrow[0,1]$ with $X([0,1]) = [0,1]$ but $P[X=0]=1$, so $X$ does not have a continuous CDF. Take the Lebesgue measure on $[0,1]$ as the probability measure, take any set $A\subseteq [0,1]$ with $P[A]=0$ but for which there is a bijective $h:A\rightarrow [0,1]$. Define $$X(\omega) = \left{\begin{array}{cc} h(\omega) & \mbox{ if $\omega \in A$} \ 0 & \mbox{ else} \end{array}\right.$$ – Michael Jul 11 '21 at 01:09
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    https://en.wikipedia.org/wiki/Cantor_set. – Oscar Lanzi Jul 11 '21 at 01:15

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