Why for perfect reflections a surface must have G2 continuity? Why for perfect reflections a surface must have G2 continuity (class A surface)?

I would like a mathematical answer.
What you see on reflects is the n-continuity of normals, which are the derivative of positions. -> a G1-only surface would have G0-only normal field, i.e.,  with sudden change of gradient in the normals (and thus, the reflects), that the eyes can notice. G2 surfaces have G1 normals fields, which is smooth enough for your eyes.