Question: How can any stochastic phenomenon follow a continuous distribution, since the probability of observing any finite set of outcomes is zero?
Answer: It’s continuous precisely because each probability is 0.
But any interval has an infinite number of points, and every interval has a probability.
This is not so for categorical distributions.
For instance, the probability that I am exactly 175.0000000… cm tall is 0. But the probability that I am between 175 and 176 cm, or 175 and 175.0001 cm is greater than 0 and that is true no matter how small the interval, at least until we get down to subatomic particle length.
On the other hand, the probability that a regular die will land on 1 is 1/6, but the probability that it will land between 1.1 and 1.9 is 0.