|Keep It Simple, Stupid|
No such assumption is made in the problem.
With a continuous probability distribution you can have some probability of picking a number in any range at all. The probability may be low, but it is always positive. That is why I said "non-zero density everywhere" in the details.
More carefully stated the experiment is defined as, "I hand you one of my two numbers at random, you look at it, hand it back, and make a guess as to high or low." The assertion is that there is a method you can use such that, no matter what my numbers are, you will have better than even odds of being right. What your odds are will depend, of course, on what my numbers are. All that you can guarantee is better than even.
Of course any purely mechanical computer will have the usual problems with "randomly pick out of a continuous distribution" however you can specify a concrete algorithm for doing that with coin-flipping (note that you don't need to find the number precisely, just determine it to sufficient detail to compare it with the one you were handed). You need make no assumptions on my numbers. And an outside observer who knows both my numbers and your method can calculate the exact probability you will be right - and that will be better than even.
(Yes, this skirts on the boundary of a lot of interesting problems, philosophical and otherwise, in math.)
In reply to RE (tilly) 1: OK, understood, but still 1 *huge* flaw