 Solution

A morphism of algebraic varieties is a map regular in local affine coordinates: for each point of the source, choose affine charts containing it and its image, and require each target coordinate to pull back locally to a quotient of polynomials with a denominator nonzero at that point. The map is continuous in the Zariski topology and pulls back regular functions on every target open set to regular functions on its inverse image.

Equivalently, one can take Zariski continuity and this local regular-function pullback property as the definition. In particular, a morphism between affine varieties is specified by globally regular coordinate functions, with values satisfying the defining equations of the target. Being an arbitrary set-theoretic function, or being merely Zariski continuous, is not sufficient.