The exchange of the volume of a region of the electric double layer of a mica surface immersed in aqueous solutions, with a dielectric constant epsilon(DL), by a nanosized radius tip, with a dielectric constant epsilon(Tip), is responsible for the repulsion at large distances from the surface (starting at similar to 100 nm, diffuse layer) and followed by an attraction when the tip is immersed in the inner layer (similar to 10 nm). The calculated dielectric constant as a function of the distance to the charged interface obtained by fitting the force versus distance curves, allows the mapping of the inner layer dielectric constant profiles with a nanometer resolution.