Amphifunctional double layers are defined by the coupling of electronic and ionic surface charging processes. They may be present at interfaces of the type metal \ oxide layer \ solution or semiconducting oxide \ solution. In the present paper, the simultaneous effects of charge-determining ions H+/OH- and specifically adsorbing ions from the background electrolyte are analyzed on the basis of the amphifunctional double layer model we developed earlier. Depending on the nature of the substrate the interactions are either or both of the type (i) ions-ionized functional surface groups (ii) ions-metallic surface atoms. For each category, the dependences of the points of zero-charge and the isoelectric point, as defined by potential-pH relationships, on the background electrolyte concentration and the individual ion binding constants are obtained. The analysis of cases (i) and (ii) allows the prediction of the shifts of iep and pzc for relaxed oxide \ electrolyte interfaces and the familiar polarized mercury \ electrolyte interface in the presence of specifically interacting ions. Attention is paid to the dependences of the ion adsorption isotherms on pH and externally applied potential. The specific adsorption of anions on TiO2 is treated in some detail: the computations presented here successfully explain experimental data.