I recently learned that for natural numbers, the Cantor Pairing function allows one to output a unique natural number from any combination of two natural numbers. According to wikipedia, it is a computable bijection $$f : \mathbb N \times \mathbb N \rightarrow \mathbb N$$ $$f(x,y) := \frac 12 (x+y)(x+y+1)+y$$ Will it generate a unique value for all real (non-integer) number values of $x$ and $y$? I believe there is no inverse function if using non-integer inputs, but I just want to know if the output $f(x,y)$ will still be unique.
Please forgive me if this isn't a worthwhile question, I do not have a mathematics background.
Edit: I'm interested in the case where we constrain $x$ and $y$ to real numbers $>0$.