So I understand how for a continuous random variable the probability of an exact value of X is zero, but then what is the value of f(x) if it's not a probability? I thought it was a probability similar to how the pmf for a discrete random variable was a piece-wise function that gave the probability for various values of X. But it can't be a probability because the function f(x) DOES take on a value for every single value of x. You plug in an x and out pops an f(x). If f(x) is indeed a probability then doesn't this contradict the idea that the probability for any single value of x is zero. So what is f(x)?