Is my random variable discrete or continuous? Suppose I have a discrete set of (finite) data (values are only positive integers) (which always remains discrete whenever the observation/survey is taken), divided into some identical categories . Now, I want to calculate the mean values of some random variable for each category, hence getting a sequence of mean values. Considering mean as a random variable (let's call it $M$), it's taking decimal values also (obviously possible). But I am not sure if $M$ is a discrete or a continuous random variable. 
My intuition suggest me that it should be discrete. Since, my original data from which I calculated the sets of mean is discrete so the mean can take finitely many values which I suppose is not the case with a continuous random variable where the mean can take infinitely many values within an interval. So is my intuition right or wrong?  Is it true that a discrete random variable takes finitely many values and continuous R.V. can take infinitely many possible values? 
 A: It is in the nature of a continuous variable, that it can take infinite number of values within any given range. In reality, this is almost never the case. If you measure the length of a board, you measure in meters, centimeters, millimeters, whatever. These are rounded values. An astronomer rounds to lightyears, a carpenter to millimeters, an machinist to tens of millimeters, but every measurement is discrete in the sense, that it is always rounded to some precision. Nevertheless, we use mathematical methods on these values as if they were continuous. As long as there are enough discrete steps so that discreteness makes no difference.
As to statistics: Psychologists often compute statistics from item counts, as if these were continuous (like the mean of a sum score of a likert scale).
Continous in this sense is not a boolean value. The question is, whether your mean is "continuous enough" to justify quasi-continuous computations.
A: If you have a finite number of samples then your intuition is correct that the sample mean cannot take certain values. You mention that the discrete values are integers so clearly their mean cannot be irrational.
We know from the central limit theory that the sample mean tends to a continuous distribution as the sample size becomes large.
Even though a finite sample size doesn't technically create a continuous distribution, if your sample size is large or if the distance between discrete values is small for your purposes then you really should consider modelling it as a continuous variable.
A: Addressing a minor side point: "Is it true that a discrete random variable takes finitely many values?"
Not necessarily. Consider the random variable "number of times that I toss a coin before it comes up heads". This is discrete but has infinitely many possible values.
