For example, suppose I am told that 10 data points come IID from a normal distribution with some mean and variance. Isn't the probability of realizing each of these values zero? Shouldn't the fact that the probability of drawing each data point being zero imply that the likelihood is zero? Why can I sample particular values rather than being forced to sample intervals, for example?
I understand that simulating draws from continuous random variables with a computer is a useful fiction, since no computer has infinite precision. However, sometimes problems are posed such that data points actually come from a continuous distribution, rather than a discrete approximation to a continuous distribution.
This seems logically impossible, or at least the zero probability should be reflected in the likelihood calculations. There is much commentary in intro probability courses about continuous RVs taking scalar values with zero probability but then this is never mentioned in a statistics class when you are told that data is IID from a continuous distribution.
I know the question is simple but I haven't seen a satisfactory answer anywhere.