  We discuss the continuum limit of discrete Dirac operators on the square
lattice in $\mathbb R^2$ as the mesh size tends to zero. To this end, we
propose the most natural and simplest embedding of $\ell^2(\mathbb Z_h^d)$ into
$L^2(\mathbb R^d)$, which enables us to compare the discrete Dirac operators
with the continuum Dirac operators in the same Hilbert space $L^2(\mathbb
R^2)^2$. In particular, we prove that the discrete Dirac operators converge to
the continuum Dirac operators in the strong resolvent sense. Potentials are
assumed to be bounded and uniformly continuous functions on $\mathbb R^2$ and
allowed to be complex matrix-valued. We also prove that the discrete Dirac
operators do not converge to the continuum Dirac operators in the norm
resolvent sense. This is closely related to the observation that the Liouville
theorem does not hold in discrete complex analysis.
