Before revision:  In the second approach ^d, on which any function in C(\mathbb{R}^d} , given any fixed f \in C(\mathcal{X \mathcal{P}_1( \mathcal{Y)) , we build non-trivial subsets of \mathcal{X} on which f }  can be efficiently approximated.
After revision:  In the second approach ^d, on which any function in C(\mathbb{R}^d} , given any fixed f \in C(\mathcal{X \mathcal{P}_1( )) , we build non-trivial subsets of \mathcal{X} on which f } can be efficiently approximated.