Analysis of symmetric matrix valued functions I

For any symmetric function f:Rn⟶Rnf:Rn⟶Rn, one can define a corresponding function on the space of n×nn×n real symmetric matrices by applying ff to the eigenvalues of the spectral decomposition. We show that this matrix valued function inherits from ff the properties of continuity, Lipschitz continuity, strict continuity, directional differentiability, Frechet differentiability, continuous differentiability.