On quasi-regulous functions

Abstract Let X be a real algebraic set. In the literature, a (real-valued) function $$\varphi $$ 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 $$f^2=\varphi ^2$$ for some regulous function $$\varphi $$ 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.