Consider a dart board that is represented by a unit circle centred at the origin. Each dart lands at a singular point within the circle (or on its outer edges). Arguments that the probability of the dart landing at a single point is $0$ are often unconvincing to me. It is clear that the probability must be strictly less than any positive real number: the probability cannot be $0.01$, for instance, as there are more than $100$ points on the dartboard, and so the sum of the probabilities would be greater than $1$. However, I still find this explanation unsatisfying, as $0 \times \infty$ is undefined (it does not equal $1$). Moreover, there is something deeply counter-intuitive about an event having a probability of $0$. When I see an event with probability $\frac{1}{5}$, I know that it means that if I repeat the experiment $5$ times, on average the number of successful trials should equal $1$. If something had a probability of $0$, then because $0$ multiplied by any number is still $0$, then no matter how many times I repeat the experiment, a successful trial would occur $0$ times on average. This seems very close to my conception of 'impossible', but perhaps because this is only an average, there is some nuance here which I am missing.
Thanks for reading.
Tl;dr: I think your question is a good one- so much so, that it will take a lot of learning about the theory behind probability before you yourself will be somewhat satisfied with an answer. (Or perhaps this is me projecting my lack of knowledge in this area).
To be honest, at first I wrote a long answer, but then I decided I don’t understand all the definitions of the terms when it comes to probability (density) functions, and I know next to nothing when it comes to The Lebesgue Theory.
However, I think (although I could be wrong) that these two links are a good start to try to improve one’s understanding of what’s going on:
https://en.m.wikipedia.org/wiki/Probability_density_function (The first few paragraphs in particular)
The sum of an uncountable number of positive numbers
However, do not take this as a full answer because I reckon there is a lot more (mathematically) to OP’s question than meets the eye- probably something to do with “probability measures / measurable functions” and “Borel Sets” (basically the Lebesgue Theory) and what have you- and I know next to nothing about these, so please someone else give a proper answer. But I can see that these things are explored towards the end of Rudin’s PMA - I just haven’t got there yet. But I thought I should share my 2 cents anyway with the above links.