  We pursue Ginsparg and Wilsons' block spin approach in the derivation of the
Ginsparg-Wilson relation and study the correspondence of the eigenmodes of the
Dirac operators in the continuum and lattice theories. After introducing a
suitable cut-off in the continuum theory, we identify unphysical modes of the
lattice Dirac operator which do not correspond to any physical modes of the
regulated continuum Dirac operator. We also consider zero modes in the
continuum and lattice theories. Our studies give a physical interpretation of
the expression of the index defined on a lattice and a formal argument on the
relation of the indices between the continuum and lattice theories.
