Families of meromorphic functions avoiding continuous functions

Leth ₁,h ₂ andh ₃ be continuous functions from the unit disk D into the Riemann sphereC such thath _i(z) ≠ h_j(z) (i ≠ j) for eachz∈D. We prove that the setF of all functionsf meromorphic on D such thatf(z)≠h _j (z) for allz ∈ D andj=1,2,3 is a normal family. The result and the method of the proof extend to quasimeromorphic functions in higher dimensions as well. The second author was supported by a Heisenberg fellowship of the DFG. The fourth author was partially supported by the Marsden Fund, New Zealand. This research was completed while the authors were attending a conference at Mathematisches Forschungsinstitut Oberwolfach in Germany. The authors would like to express their sincere thanks to the Institute for providing a stimulating atmosphere and for its kind hospitality.