My guess would be discrete because such a variable would only take a countable set of values and "continuous" seems to imply the continuum of the real numbers.
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1It won't count as continuous, since you have points with non-zero probability. Whether it counts as discrete or not depends on exactly whose definition of discrete you use (there are other possibilities than discrete/continuous). By the definition I was taught, the answer is "yes that's discrete", but by another common definition the answer is "no". This doesn't change anything material other than the label we apply of course. – Glen_b Sep 21 '15 at 01:16
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1A great answer specifying exactly what the pmf of such a random variable might look like is [here](http://stats.stackexchange.com/a/104018/6633) – Dilip Sarwate Sep 21 '15 at 02:19
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These are both very helpful, clear responses. – user89964 Sep 21 '15 at 03:53
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A continuous variable has positive length, area or volume. These terms, respectively, correspond to particular Lebesgue measures. But the Lebesgue measure of $\mathbb{Q}$ is 0; it doesn't have positive length.
Approached from another perspective, it is sufficient to note that $\mathbb{Q}$ is countable, so it's not continuous.
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