Why is it P(X=a) = 0 for all a belonging to ℝ for continuus rand.vars?

[cdux]

I probably miss something basic, unless it's an abstract definition.

 

[HallsofIvy]

$$P(X\in [a, b])= \int_a^b f(x)dx$$
where f(x) is the "probability density function".

P(X= a) means $$P(X \in [a,a])= \int_a^a f(x)dx= 0$$

 

[cdux]

Thank you. I guess it's like my initial assumption: "it represents an atomic elelement of the integral and that tends to 0".