In this talk we will consider the homogeneous function space BMO$(\Omega)$ and the inhomogeneous function space bmo($\Omega$). We are interested in characterizing the extension domains of both function spaces. Several such characterizations are possible and their equivalence in the BMO setting was studied by Gehring and Osgood. We will revisit two of their theorems and adapt them to the inhomogeneous context. This is joint work with Galia Dafni.