If we think of z = F(x, y) as describing a surface, then this question translates to asking:
Can one find a continuous surface for which every distinct point on that surface has a unique height
Clearly not. There can only be two points with unique height, namely the global maximum and minimum. The rest must either have an infinite number of solutions (a curve) or none.
Argument: By projecting R^2 onto R, you lose information that you cannot retrieve.
If that argument is valid, then it's validity must depend on the nature of the real number, not the fact that you lose information as R has less dimension than R^2. For example, there exists the Cantor pairing function which uniquely maps N^2 to N, and can be inverted.
Furthermore, here's a method for storing the information of a pair of real numbers into a single one: Suppose we have to encode the two numbers with decimal (any sensible base should work) representation
a1 a2 a3 . a4 a5
b1 b2 . b3 b4 b5
Then, we can interleave each digit (and pad with 0 to make the decimal point
line up) to form the resulting z to be:
a1 0 a2 b1 a3 b2 . a4 b3 a5 b4 0 b5
And thus, one can restore any number a and b from that combination.
The translation of the question in terms of surfaces is actually wrong, because it requires continuity, whilst finding a bijection doesn't.
It is possible to uniquely encode two real numbers from into one real number (and thus have an injection). There are some complications I was a bit lazy to learn about when trying to make it a surjection, but it seems possible (do you encode .5 as .499999999... or .50000000...?). I'm confident to assume that it's possible because
- some of the sources I've seen say it is
- I'm pretty confident one could develop a grammar to have "escape sequences", except you'd be restricted to digits instead of ascii characters (for example, agree that 00 means the next number is special. If it's a 1, then apply a certain rule to support an edge case, etc... And if you wanted an actual 00, then that's just another edge case). This is analogous to backslashes in many programing languages.
However, from the "visual" argument (with surfaces), it now seems clear to me that one cannot find a continuous bijection.
Credit to the MATH1241 lecture for the question and ideas (and of course the internet). Mistakes (and obvious lack of rigorousness) are mine.
Here are some of the resources I kept track of (obviously not very reliable, but come on, I don't even know what continuity means, I'll call this good enough for me, for now).
https://www.reddit.com/r/askscience/comments/6opvae/simplest_proof_that_r2_r/
[I obviously used a bunch of other ones, google's your friend, but I think this ones are pretty good]