If -~<α<β<~ let mid(α,x,β)=α if x<α, x if α=<x=<β, β if x>β. Let A n =B n +P n where B n and P n are nxn Hermitian matrices. We show that if P n F 2 =o(n) then, for any [α,β], (A) i = 1 n |F(mid(α,λ i (A n ),β))-F(mid(α,λ i (B n ),β))|=o(n) if F C[α,β]. (Eigenvalues numbered in nondecreasing order.) We consider the special case where {P n } are real Hankel matrices. We also show that if rank(P n )=o(n) then (A) holds for every [α,β] and F C[α,β]. Combining these results yields a result concerning C n =B n +E n +R n , where E n 2 F =o(n) and rank(R n )=o(n). We also consider the case where the conditions on {E n } are stated in terms of Schatten p-norms. Finally, we show that if {T n } are Hermitian Toeplitz matrices generated by f C[-π,π] with minimum m f and maximum M f , (2(i-1)-n)π/n=<ξ i n =<(2i-n)π/n, 1=<i=<n, and τ n is a permutation of {1,2,...,n} such that f(ξ τ n ( 1 ) , n )=<f(ξ τ n ( 2 ) , n )=<...=<f(ξ τ n ( n ) , n ), then i = 1 n |F(λ i (T n ))-F(f(ξ τ n ( i ) , n ))|=o(n) if F C[m f ,M f ].