[jason_t]

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I've never taken probability. Undergraduate or otherwise.

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In all seriousness, I think the misunderstanding lies here. You don't know what a continuous-valued random variable is, or what a probability density function is (and why it is different from a probability mass function). It may be non-intuitive, but a continuous-valued RV may take on a particular value, even though the probability of that event is zero (although, I believe, the pdf must be nonzero).

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I know what a random variable is. (It's just a measurable function, right? Maybe we want {x: f(x) = +-oo} to have zero measure too.)