Why is "the probability that a continuous random variable equals some value always zero"? 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 $X\sim N(0,1)$, 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
 A: 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.  
A: Let's just go with a simple intuitive explanation. But it can get really mathy really fast (if you prefer).
A continuous distribution is a line between two points A and B. On this line there are infinitely many points, no matter if the distance between the points (A, B) is extremely small. If all of those infinite points had a probability larger than 0, then the sum of probabilities would be infinity. 
But if that where the case, then the said probability distribution would violate the (Kolomogorov) axioms of probability. So it would not be measure of probability in the modern understanding. 
Edit, the derivative of the function $\text{min}(x,y)$ is not defined for $x=y$. So your example is not continuous. 
