A class of universal approximators of real continuous functions revisited

We revisit the theorem stating that it is possible to approximate with any accuracy any real continuous function with a class of relational maps. In other words, relational maps are universal approximators. We review the key works that have proved this property, highlighting their limitations and providing yet another proof that it is not restricted by certain assumptions considered in early proofs. We also show how one can go inversely to approximate these systems with a series of polynomials. This provides us with analytical expressions of these maps which can facilitate a series of important analysis tasks such as modeling and numerical analysis of ill-defined-uncertain complex systems.