Although the classification of affine cubic curves was undertaken by Newton(4), in one of the first major exercises ever in coordinate geometry (see Cayley(2) for a fuller account), a parallel study of cubic functions seems not to have been contemplated till recently. The essential difference is that a function f defines a pencil of curves – its level curves – which have to be considered simultaneously. The author's interest in the subject arose from problems in singularity theory (concerning canonical stratifications), and a later paper in the series will have applications of this kind. Here we study the simplest case as an introduction. Our techniques are entirely classical, but the results are hard to find elsewhere. In later papers we intend to study cubic functions on ℝ2, and on ℂ3.