Differentiability properties of rotationally invariant functions

Let f be a function on the set Lin of all tensors (= square matrices) on a vector space of arbitrary dimension. If f is rotationally invariant (with respect to the left and right multiplication by proper orthogonal tensors), it has a representation (f) over tilde through a symmetric even function of the signed singular values of the tensor argument A is an element of Lin. It is shown that f is of class C-r,r=0,1,...,infinity, if and only if (f) over tilde is of class C-r, and an inductive formula is given for the derivatives D-r f.