My attempt at explaining surface continuity.
it really boils down to to:
g0, no continuity
g1, tangent continuity, no abrupt change in surface direction but abrupt change in radius
g2, gradual change not only in surface direction but also change of radius is smooth and gradual
g3, Not only is radius change gradual but the rate of that change is gradual

Practical difference in reflection impact is the crucial point why these matter. I tried to demonstrate this with ray tracing and showing the way the reflections behave in these different surface transitions.
if you see any errors please comment.

(I know I called flat surface to have radius of zero and the gradually growing when I meant infinite and gradually getting smaller, later I sorta said it right)