Algebraic versions of Hartogs' theorem

Let K be an uncountable field of characteristic 0. For a given functionf:Kn -> K,withn >= 2, we prove thatfis regular if and only if the restriction f |C is a regularfunction for every algebraic curve C in Kn which is either an affine line or is isomorphictoaplanecurveinK2defined by the equation Xp-Yq=0, where p<q are prime numbers. We also show that regularity off can be verified on other algebraic curvesinKnwith desired geometric properties. Furthermore, if the field K is not algebraically closed, we construct aK-valued function on Kn that is not regular, but all its restrictions to nonsingular algebraic curves in Kn are regular functions.