0 shouldn't be assigned as a probability if you're going to do Bayesian updates. That doesn't interfere with the necessity of using 0 when assigning probabilities to continuous distributions, as any evidence you have in practice will be at a particular precision.

For example, say the time it takes to complete a task is x. You might assign a probability of 20% that the task is finished between 2.3 and 2.4 seconds, with an even distribution between. Then, the probability that it is exactly 2.35 seconds is 0; however, the measured time might be 2.3500 seconds to the precision of your timing device, whose prior probability would be .02%.

Edit: I need a linter for these comments. Where's the warning "x was declared but never used"?