  In this article we study the behavior of the Oh-Schwarz spectral invariants
under C^0-small perturbations of the Hamiltonian flow. We obtain an estimate,
which under certain assumptions, relates the spectral invariants of a
Hamiltonian to the C^0-distance of its flow from the identity. Using the
mentioned estimate we show that, unlike the Hofer norm, the spectral norm is
C^0-continuous on surfaces. We also present applications of the above results
to the theory of Calabi quasimorphisms and improve a result of Entov,
Polterovich and Py. In the final section of the paper we use our results to
answer a question of Y.-G. Oh about spectral Hamiltonian homeomorphisms.
