We show that covalent sidewall functionalization of single-wall nanotubes leads to drastic changes of nanotube electronic states near the Fermi level. The sp(3) hybridization between the functional group and nanotube induces an impurity state near the Fermi level. The impurity state is found to be extended over a large distance (> 1 nm) even though the structural deformation is confined to the vicinity of the functionalizing site. Thus, dramatic changes in the conductive properties of the nanotube can be expected even if the concentration of functionalization molecules is small. This effect provides an effective pathway for band structure engineering, nanoelectronic device, and sensor applications through covalent sidewall functionalization.