The radial structure of the shear Alfvén wave continuous spectrum is calculated inside the separatrix of a magnetic island. We find that, within a magnetic island, there is a continuous spectrum very similar to that of tokamak plasmas, where a generalized safety factor q can be defined and a wide frequency gap is formed, analogous to the ellipticity induced Alfvén eigenmode gap in tokamaks. This is due to the strong eccentricity of the island cross section. In this gap, a magnetic-island induced Alfvén eigenmode (MIAE) can exist as a bound state, essentially free of continuum damping, which can be resonantly excited by energetic particles and interact nonlinearly with the magnetic island. Because of the frequency dependence of the shear Alfvén wave continuum on the magnetic-island size, the possibility of utilizing MIAE frequency scalings as a novel magnetic-island diagnostic is also discussed.