Double-walled carbon nanotubes (DWNTs) present an original coaxial geometry in which the inner wall is naturally protected from the environment by the outer wall. Covalent functionalization is introduced here as an effective approach to investigate DWNT devices. Performed using an aryldiazonium salt, the functionalization is reversible upon thermal annealing and occurs strictly at the surface of the outer wall, leaving the inner wall essentially unaltered by the chemical bonding. Measurements on functionalized DWNT transistors show that the electrical current is carried by the inner wall and provide unambiguous identification of the metallic or semiconducting character of both walls. New insights about current saturation at high bias in DWNTs are also presented as an illustration of new experiments unlocked by the method. The wall-selectivity of the functionalization not only enables selective optical and electrical probing of the DWNTs, but it also paves the way to designing novel electronic devices in which the inner wall is used for electrical transport while the outer wall chemically interacts with the environment.