I'm totally confused about the concept of Probability Distribution Function for continuous sample space.I read that probability of an event in continuous space is zero and it seems logical since we have an infinite number of points also as quoted by Wikipedia the absolute likelihood of an event exactly occurring is zero. But when we substitute a value x in the Normal distribution function we get a finite value . Can someone explain to me whether we can use PDF for a exact value or should we consider only a range to compute the probability of a range? Also when we substitute mean in the Gaussian the value seems to be very high indicating the mode and mean are the same.
Edit: Thank you for the answers but when we write the probability of data set given the mean and variance in the maximum likelihood function, we actually use the Normal distribution PDF directly for a input vector X which is N(x|mean,variance) , here we are using the PDF as if to represent the probabilities of the input vector.