  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 article, 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. Furthermore, we obtain
that the growth of the action width of a Hamiltonian homeomorphism increases at
least linearly, and that the group of Hamiltonian homeomorphisms of
$\mathbb{T}^2$ and the group of area preserving homeomorphisms isotopic to the
identity of $\Sigma_g$ ($g>1$) are torsion free, where $\Sigma_g$ is a closed
orientated surface with genus $g$. Finally, we will show how the $C^1$-Zimmer's
conjecture on surfaces deduces from $C^0$-Schwarz's theorem.
