Families of meromorphic functions avoiding continuous functions

Let h₁, h₂ and h₃ be continuous functions from the unit disk double-struck D sign into the Riemann sphere ℂ such that h_i(z) ≠ h_j(z) (i ≠ j) for each z ∈ double-struck D sign. We prove that the set ℱ of all functions f meromorphic on double-struck D sign such that f(z) ≠ h_j(z) for all z ∈ D and j = 1, 2, 3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well.