A random variable, $ X $, is a function that assigns to each element in the sample space a unique value. The sample space is the domain of $X$. The value assigned to each element in the sample space is the probability for that particular element in the sample space occurring. The sum of the probabilities over the sample space must equal 1, and the probability that the random variable takes on a value $ x $ must be between 0 and 1, inclusive.
As an example, consider the random variable $ X $ which is defined by the function $ f(x)=1 $ on the interval $ 0 \le x \le 1. $ This is indeed a probability density function because $ 0 \leq P(X) \leq 1 $ and all the probabilities in the sample space taken together (the area) sum to 1. This particular pdf defines a uniform distribution.
This pdf defines a continuous random variable because the domain takes on all real values in the sample space. PDF's that are not continuous are discrete which means the sample space is finite. The interval $(0,1)$ is not finite.