An r-dependent term is added to the metric for the study of extensions of the Theory of General Relativity (GR). The stability of the Schwarzschild solution is investigated with and without the presence of this additional term. Polar (Zerilli) and axial (Regge-Wheeler) modes are studied and the differences discussed. The main conclusion is that, with an r-dependent term in the metric, the axial and polar modes remain stable. Assuming that GR describes the data of the gravitational wave events, in the modified theory, axial modes involve larger masses, which might change the interpretation of the source of the gravitational wave events observed.