[en] The continuous spectrum of Shear Alfven waves is calculated for finite-beta tokamak equilibrium in the presence of a finite-size magnetic island. The beta-induced Alfven Eigenmodes continuum accumulation point (BAE-CAP) is found to be positioned at the separatrix flux surface, while the frequency is not changed. The most remarkable feature is found to be the presence of new CAPs at the O-point of the island, which give rise to gaps in the continuous spectrum. This could make the existence of new magnetic-island induced Alfven Eigenmodes (MiAE) possible, excited via wave-particle resonances if the island is sufficiently wide with respect to the mode radial localization. Experimental data of modes observed in Frascati Tokamak Upgrade (FTU) are presented, which agree with theoretical scalings. Due to the frequency dependence on the magnetic island size, the feasibility of utilizing MiAE as novel magnetic island diagnostic is also discussed. (author)