Automorphism of unit disc maps unit circle onto itself

Question

If I have an automorphism of the open unit disc, then I think that it naturally extends to a holomorphic function from the closed unit disc into itself (use continuity to define values on the boundary), but I can't see why this new holomorphic function would map the unit circle onto itself? Why would points of the unit circle will map to points of the unit circle (continuity arguments?), and, more importantly, why would the image of the unit circle be all of the unit circle? I can't prove it using open mapping theorem/ other elementary arguments that come to mind. Please do not quote the automorphisms of the unit disc: I already know them, and this question actually sprung from motivating arguments for the theorem that states all automorphisms of the unit disc.