Problem formulation classification task I would like to know if it is correct for a classification task in a supervised learning to say the model we are looking for is a function from RxR to a discrete space
$$
f:\mathbb{R}\times\mathbb{R} \rightharpoonup\left\{ Label1,Label2\right\}
$$

No, I say that both sides are incorrect in general.
If you have two predictor variables, both of which can take any real value, then the left is fine. If you have three such variables, you need another $\times\mathbb R$. If your predictor variables are categorical, then I would not call them variables on all of $\mathbb R$.
For the right side, your function likely outputs probability values. Consequently, I would consider the target to be $[0,1]$, not a discrete space.
