On closed symplectically aspherical manifolds, Schwarz proved a classical result that the action function of a nontrivial Hamiltonian diffeomorphism is not constant by using Floer homology.In this talk, we generalize Schwarz's theorem to the C^0-case on closed aspherical surfaces (i.e., closed oriented surfaces with genus more than 1). As an application, we prove that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected. We also get a similar result of an area and orientation preserving homeomorphism of the 2-sphere by applying Brouwer plane translation theorem. Furthermore, we can prove a general version of C^1-Zimmer's conjecture based on the C^0-Schwarz's theorem.