  For an arbitrary function f:\Omega \rightarrow C (where \Omega is a subset of
the field C) and a positive integer k let f act on all diagonalizable complex
matrices whose all eigenvalues lie in Omega in the following way: f[P
Diag(z1,...,zk) P-1] = P Diag(f(z1),...,f(zk)) P-1 for arbitrary numbers
z1,...,zk in \Omega and an invertible k \times k matrix P. The aim of the paper
is to fully answer the question of when the function fop defined above is
continuous for fixed k. In particular, it is shown that if \Omega is open in C,
then fop is continuous for fixed k > 2 iff f is holomorphic; and if \Omega is
an interval in R and k > 2, then fop is continuous iff f is of class
Ck-2(\Omega) and f(k-2) is locally Lipschitz in \Omega. Also a full
characterization is given when the domain of f is arbitrary as well as when fop
acts on infinite-dimensional (diagonalizable) matrices.
