On quasi-regulous functions

Let X be a real algebraic set. In the literature, a (real-valued) function φ on X is called regulous if its restriction to each algebraic subset of X is a continuous rational function. A function f on X is called quasi-regulous if it is continuous and f²=φ ² for some regulous function φ on X . Assuming X is nonsingular, we prove that a function f on X is quasi-regulous if and only if its restriction to each algebraic curve in X is quasi-regulous. We also prove three other variants of this result.