Data Structure for representing several sets of continuous real numbers

You both are not quite right. The situation is more delicate.

First of all, nothing in a computer presents a continuous set. The whole computer is the finite-state machine, like it or not. You should also understand, that there are no real numbers (a set of those would be a continuous set) -- you have only the model approximating that, such as the set of all double of float numbers.

Now, the real problem is how to build a model of real number or other "potentially continues" objects to make some help from the computer. This is the completely different, quite correct problem, but only if strictly formulated.

In other words, the solution of OP's is most likely quite solvable. He only need to provide complete and strict description of required operations. Listing of those set-theory primitives is not enough. I would need to know how the set is used. I assume the use is realistic, and in this case I can almost guarantee the solution. I would recommend to start with explanation of the ultimate purpose.

--SA

Absolutely. Thank you very much.
--SA

I did not down-vote this answer even in its original version, because it is not completely wrong (and it was not completely wrong before); the question just needs some more detail and it all needs some analysis based on that.
And the use of the Range is a good idea. The set could be represented by the set of instances of, say, Range<double>.
Depending on expected use of the set class, even the modified use of HashSet could be appropriate. Let's learn about the prospective usage of it.
--SA