No, it doesn't. A continuous distribution is just a way of assigning probabilities to a parameterized set of propositions where the parameters are continuous and so the set is infinite. But for any given proposition in that set the prior is a number.
> Priors stands for prior distribution
No. Bayes's theorem is:
P(H|E) = P(E|H) * P(H) / P(E)
The "prior" is P(H) which is a probability, i.e. a number between 0 and 1, as are all the other "P" values. The "*" and "/" operations in Bayes's theorems are scalar multiplication and division.
A continuous distribution does not assign probabilities to each proposition but subsets of them. To see this concretely consider the continuous Uniform distribution from 1 to 1.5. It will for all values of its support have a probability density of 2. Most people would not consider 2 a probability.
For continuous distributions, Bayes's theorem becomes about probability densities and not probabilities.
Nope. It assigns probabilities to individual propositions.
> Most people would not consider 2 a probability.
That's true, but in your example 2 is not a probability but a probability density, and a probability density is not the same as a probability. To get a probability out of a probability density you have to integrate. For each possible interval over which you could integrate there is a corresponding proposition whose probability of being true is exactly the value of the integral.
(Note that there are all kinds of ways that statements can fail to be propositions. For example, you can't assign a Bayesian prior to the statement: "All even integers are green" despite the fact that you know what all the words mean.)