Does it make sense to compare the probability of two outcomes of a continuous pdf? Given a continuous pdf $f(x)$ and two values $a, b$ in the domain of $f$, suppose that $f(a) > f(b) > 0$. Taking into account that $P(a) = P(b) = 0$, does it make sense to claim that $a$ is still somehow more probable than b? If yes, in what sense?
Arguing intuitively, let $f(a) = 10^{10}$ and $f(b) = 10^{-10}$. If you were told that on a random drawing the outcome is one of these two values ($a$ and $b$), which value would you bet on, if you had to?
 A: A single point tells you nothing. If you had some lottery asking you to choose a real number, and they extract a real number between 0 and 1 (barring the fact that it is technically impossible, as physical misurations are all rational) then you'd never catch the target. Never. It's like saying that you have to get all digits of an infinite number right (and it is also way more difficult than that): you simply won't be able to manage to do that.
What matters is that, with continuous pdfs, you can be reasonably certain that, in the example you posted, it would make more sense to bet on $a$ than on $b$. That is because continuous pdfs are, well, continuous: there will be an interval (small, perhaps, very small) centered around $a$ that has a bigger probability than another interval (not necessarily of the same dimension) centered at $b$, as in a neighborhood of $a$ the function will have a bigger value than in a neighborhood of $b$. So your money should be on an interval around $a$, if you know that the pdf is continuous.
TL,DR: the value of a pdf in a point tells you nothing, unless you know that the pdf is continuous at that point, and that is an information about neighborhoods of the point.
A: In terms of applications, I do think there is an intuitive sense that one point is more probable than the other.  Maybe, from pure math standpoint, of course the point itself has zero probability, so it makes no sense to compare.  But in general life, you will really care about either a region around the point or the distribution above or below that point. 
Consider example of oil price in DEC2018.  There is a very active futures market for WTI.  It is literally a Bayesian betting situation.  From the pricing of different puts and calls, you can extract the probability distribution curve.  EIA publishes a CI "funnel", but there is even more detail implicit.  (See link below.)  So, if I see a news story saying that $50 is more likely than $100, I know what they mean.  [Either something like a plus or minus $ region along the curve, or (for a hedger), the region above/below.]
https://www.eia.gov/outlooks/steo/report/prices.cfm (see the first figure)
Actually even for a pure math comparison, I don't see why you can't make a comparison between the two as some sort of L'Hopital ratio (as the region around each point gets smaller).  I mean think about comparison of rainfall accumulation at a literal point in the desert versus a point in the jungle.  Yes, the actual amount of water is zero in either case, but we just get around that by talking about a height of water (inches) versus a volume.  And it still has day to day relevance to know which is dryer or wetter.   
