regular and rational functions

Two kinds of functions live on a variety, and the difference is whether dividing is allowed. A regular function is one given, locally, by an honest polynomial — no denominators, defined everywhere on its open patch and behaving like a smooth quantity. A rational function is allowed a denominator, so it may be a perfectly good function on most of the variety but blow up or become undefined on a thin closed set. Regular is to rational as a polynomial is to a fraction.

Make this local. A function f on an open set U of a variety X is regular at a point p if near p it can be written as a ratio g/h of regular functions on the ambient space with h(p) not zero; on an affine variety the globally regular functions are exactly the elements of the coordinate ring k[X]. A rational function is an equivalence class of pairs (U, f) with f regular on a dense open U, two pairs identified when they agree on the overlap; these form the function field k(X). The crucial subtlety is the domain of definition: a rational function may have several representations as g/h, and it counts as regular at p if any one of them has nonvanishing denominator there, which can rescue points a single formula seems to forbid.

Together these notions make precise what a 'function' on a variety is and underpin morphisms and rational maps. The classic surprise: on a smooth projective curve the local pictures glue so well that a rational function regular at every point must be constant, and more generally the only regular functions on any irreducible projective variety are the constants. Caveat: never read off a rational function's domain from one formula. The function (x^2 - y^2)/(x - y) looks undefined on x = y but equals x + y wherever both are defined, hence extends regularly across that line — the apparent singularity was an artifact of the chosen representative.