Dear all, as we know, principle curvature $\displaystyle k_1, k_2$ (the eigenvalue of the Weingarten transform) is a continous function near an umbilical point. However, I could not give an example showing that this could not be improved (to be a smooth function!)

Thank you very much, and would you show me an example --- that means, a surface with the principle curvature being exactly a continous function, but not a differential one?