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. Our methods involve the theory of transverse foliations for dynamical systems of surfaces inspired by Le Calvez and its recent progresses. 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 preserving and orientation preserving homeomorphism of the two sphere by applying Brouwer plane translation theorem. In the end, we will give further applications that we obtained recently based on the $C^0$-Schwarz theorem.