I was reading up on discrete random variables, when I came across the definition:

A random variable is called discrete if its range (the set of values that it can take) is finite or at mostcountably infinite. A random variable that can take anuncountably infinitenumber of values is not discrete. For an example, consider the experiment of choosing a point a from the interval [−1, 1]. The random variable that associates the numerical value a^2 to the outcome a is not discrete.

I do not quite understand the meaning, even after looking at other definitions which talked about a 1:1 correspondence with positive integers.

Does this mean that (1,2], [0,3] are countably infinite, whereas [-2,2] or (-3,2] are not, as there are frequently two numbers in the set which will correspond to the same positive integer? (eg -1.5^2 and 1.5^2 both correspond to 2.25)?

additionally,

What does the notation sgn(a) = { 1 if a>0, 0 if a=0, -1 if a<0

imply? I have not seen 'sgn' before

thanks