continuous and discrete random variables According to my understanding regarding discrete and continuous random variables is that discrete random variables are variables that can take finite values that have resulted from a finite number of outcomes or if the values are infinite but still countable ( whole numbers) then the variable is also classified as a discrete random variable. On the other hand , if the random variable can take any value from an infinite outcomes set or if it can be assigned any value from a finite interval ( whole and decimal numbers so that the possible values are infinite ) then it is called continuous random variable. Now for a certain random experiment , let the set of possible values of a random variable to be {1 , 1.2 , 1.2222 ,1.2554 , 2 , 2.54 ,2.66666 ,3} , then the random variable is classified as a discrete or continuous random variable ? I am confused here , since this sample space is finite and hence it is discrete random variable , however , the possible values include whole and decimal numbers so can it be classified as a continuous random variable ?

It depends whether the set of possible outcomes $\Omega$ is countable (which includes some infinite sets) or uncountable. In the former case, you have a discrete random variable (RV), in the latter case a continuous one.
For the geometric distribution with $P(X=m)=p(1-p)^{m-1}$, e.g., the set of possible values $m=1,2,\ldots$ is infinite, but only countably infinite and it is thus a discrete RV with $\sum_k P(X=k)=1$. This is an example of countably infinte set, which means that there is a bijective mapping between the set and the natural numbers.
A finite interval of real values, in conmtrast, is an uncountable set because there are uncountable possible values in between. It can be proven that such a set is uncountably infinite, i.e., there is no bijective mapping onto the natural numbers.
In case of interest, please have look at Cantor's set theory (the link is to an informal introduction).
