On a real algebraic variety a continuous function can be rational along every algebraic curve, or along every analytic arc, without being rational on the variety itself. This Idea develops those two regularity classes, the finer notion of a hereditarily rational function, and the criteria that force such a function to be genuinely regular or rational: on a smooth variety curve-rational and arc-rational functions coincide with continuous rational ones, while on singular varieties the classes separate. Students learn to test a continuous function by restricting it to curves and arcs, and to see why the singular locus is exactly where the equivalences break down.
On a real algebraic variety X, a function on a subset of the real points X(R) is regular if it is locally a quotient p/q of polynomials with q non-vanishing, and rational if it agrees with a regular …