Local interface response effects are neglected based on the traditional electromagnetic boundary conditions (EMBCs) in an abrupt interface model. In this study, generalized nanoscale EMBCs are derived with interface response functions (IRFs) representing field inhomogeneity across the interface based on integral Maxwell's equations. They are rewritten in two different forms that correspond to the equivalent abrupt interface models with interface-induced dipoles or charges and currents. Interesting behaviors of Brewster angle shifting, non-extinction at Brewster angle, and unique absorption or gain effects are revealed based on the advanced Fresnel formula. IRFs-controlled GH-shift and angular GH-shift of a Gaussian beam near the Brewster angles are generated by the gradient interface. These unique phenomena provide some guidance for measuring the IRFs and expanding interface photonics at the nanoscale.