They absolutely do define gradients via functions - but the point it, the screen only has a finite number of pixels. As I understood it, pixels is what this technique uses as it's dicing primitive. Pixel sized rectangles cut out of the polygon.

Pixar's Renderman used to use a (slightly) related technique. This was along time ago, and I have no idea if they still do things this way - but they would subdivide patches until each primitive was smaller than a pixel.