Consider the famous two-envelopes problem, summarized below:
A wealthy eccentric places two envelopes in front of you. She tells you that both envelopes contain money, and that one contains twice as much as the other, but she does not tell you which is which. You are allowed to choose one envelope, and to keep all the money you find inside.
Now let $p(x)$ be the PMF for the value in the smaller of the envelopes. If the value of the envelope you chose was $a$, it's easy to see that the probability the envelope you choose is the smaller of the two envelopes is:
$$\frac{p(a)}{p(a/2) + p(a)}$$
Now in the continuous case, let $f(x)$ be the analogous PDF for value in the smaller of the two envelopes. As seen in https://stats.stackexchange.com/questions/506499/computing-probability-distributions-in-the-two-envelope-problem, now you have to be a bit careful with the PDF transformation, and in the continuous case the probability you chose the smaller of the two turns out to be
$$\frac{2f(a)}{f(a/2) + 2f(a)}$$
But this brings about something puzzling. In the discrete case, if you asusme that $p(a/2) = p(a)$ then the probability you hold the smaller of the two envelopes is 1/2, as we might expect.
But in the continuous case, if $f(a) = f(a/2)$ now the probability we hold the smaller of the two is 2/3. Why does switching the setting from a discrete to a continuous one suddenly increase our confidence that we hold the smaller envelope, despite both cases using the same "flat" prior (flat for a given value of $a$).
If you follow the actual calculation for how the probability in the continuous case was computed, you can see the factor of $2$ in $2f(a)$ comes from the fact that you consider the PDF at $a/2$ around an interval $da/2$, while considering the PDF at $a$ is around $da$. But given that the $da$ cancel out when taking the ratio, it's hard to intuitively see why this should make a difference when considering the probability that we hold the smaller envelope.
Similarly, given that we can always use a series of dirac deltas to create a PDF from a given PMF which should always give the same answers, it's again hard to reconcile the differences.