@humy said
@Soothfast
Although that doesn't answer my question, I appreciate your input.
I tried but so far have failed to find any example of a conventionally defined continuous distribution where it can make sense for probability density of some specific value of x equaling +infinity.
I have also tried but so far have failed to CONTRIVE a hypothetical continuous distribution with ...[text shortened]... ssarily doesn't make sense but I have been so far been unable to pin down WHY thus I could be wrong.
The derivative of the function F_X I constructed, which I denote by f_X, is
f_X(x) = 0, for x < -1
f_X(x) = (1/6)x^(-2/3), for -1 <= x < 1, with f_X(0)=+∞
f_X(x) = 0, for x >= 1
Integrate this from -∞ to +∞ and you get 1, as required. It is a probability density function (p.d.f.) as near as I can tell, and it gives the probability density at x=0 as being +∞. (You technically should determine f_X(0), or F'_X(0), using the limit definition of derivative to see this.) The random variable X is continuous since F_X is continuous everywhere, and f_X (i.e. F'_X) exists and is continuous everywhere except at -1, 0, and +1.
Elementary stat books might say a p.d.f. f_X has to be everywhere continuous, but in more advanced treatments this requirement is relaxed a bit, so that for instance a finite number of discontinuities may be present. Above, f_X has jump discontinuities at -1 and +1, and an infinite discontinuity at 0. I think it might be an example of what you're looking for, though I'm not sure what you mean by a "conventionally defined" continuous distribution.
In the real world something like charge density goes to infinity at the precise location of an electron (never mind the Uncertainty Principle here). But an electron is classically treated as a point in space, I believe? To compute total charge in some region containing an electron you would be integrating over the region, including the place where the electron is. Nevertheless you don't find the total charge to be infinite.
Also look here, it caught my eye:
https://tinyurl.com/y5kmnpas