I found lots of references that say, "the probability that a continuous random variable equals some single value is always zero". Why is that?

Here is a counterexample I thought of: supposing $Xsim N(0,1)$, define $Y=min(X,0)$. Then Y is a continuous random variable but the probability of $Y$ at a single point $0$ should be $0.5$, not zero.

Also, I think any CDF would be left continuous if "the probability that a continuous random variable equals some single value is always zero".

What is wrong with my thoughts?

P.S. Examples of the references are:

- http://www.henry.k12.ga.us/ugh/apstat/chapternotes/7supplement.html
- http://mathinsight.org/probability_distribution_idea

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#### Best Answer

The thing is $Y$ is not that continuous to begin with. To be continuous, the distribution function of $Y$ must be absolutely continuous (see definition 1.32, page 10 of link http://math.arizona.edu/~jwatkins/probnotes.pdf by @fcop). You see the distribution of Y has a half impulse (Dirac delta) function at zero. When you approach zero on the negative side there is a jump in distribution function value. So the distribution function of $Y$ is not continuous.

If $f(x)$ is continuous, $g(f(x))$ is not necessarily continuous.

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