Probability vs Density

Part Two: The PMF that wants to be PDFURL copied

Let's brute force our way through a problem of infinite possibilities. A simple example would be a random real number generator between 0 and 1/2. We generate 10000 numbers and bucket them into intervals of fixed width. E.g. We can see that roughly a fifth of the numbers would be between each 0.1 bucket But nothing stops us from fixing the bucket size, so we could make it smaller and smaller. As you can see, the heights get smaller and smaller since the probability of a number falling in a smaller bucket is small. And here is the heart of the problem. The PMF can never work for continuous distributions because it is guaranteed to give 0 probabilities for every bucket at infinite precision.
But continuous distributions don't care about the resolution of a histogram. We both know our random-number-generator has a probability distribution. We just don't know what it is. We need a better tool to map a continous distribution.