Let A andà be n × n diagonalizable matrices and f be a function defined on their spectra. In the present paper, bounds for the norm of f (A) − f (Ã) are established. Applications to differential equations are also discussed. http://math.technion.ac.il/iic/ela whose solutions are u(t) andũ(t), respectively. Let us take the initial conditions u(0) =ũ(0) ∈ C n . Then we have u(t) −ũ(t) = (exp [At] − exp [Ãt])u(0). By (1.2), u(t) −ũ(t) ≤ u(0) N 2 (exp [At] − exp [Ãt])