For example, take any function z=f(x) that has only 2 order derivatives, that is f is C^2 but not C^3.

Try to compute the principle curvature of a surface parametrized by (x,y) -> ( x, y, f(x) )

k1 is a function involving f'' and f', k2 is 0.

Since f'' is not differentiable, k1 is only continuous.