nLab
rational function

Given a commutative ring $R$, the commutative ring of rational functions with coefficients in $R$ is the field of fractions of the polynomial ring $R[z]$.

Let $X$ be an affine variety over a field $k$ with the ring of regular function?s $𝒪(X)$. A rational function is any element of the field of fractions of $𝒪(X)$, that is the function field of the variety.

In either case, rational functions are equivalence classes of fractions; they need not be functions defined everywhere.