  One of the most natural and challenging issues in discrete complex analysis
is to prove the convergence of discrete holomorphic functions to their
continuous counterparts.
  This article is to solve the open problem in the general setting. To this end
we introduce new concepts of discrete surface measure and discrete outer normal
vector and establish the discrete Cauchy-Pompeiu integral formula,
  \begin{eqnarray*} f(\zeta)=\displaystyle{\int_{\partial B^h}}
\mathcal{K}^h(z,\zeta) f(z)dS^h(z)+\displaystyle{\int_{B^h}} E^h(\zeta-z)
\partial_{\bar z}^h f (z)dV^h(z),\end{eqnarray*} which results in the uniform
convergence of the scaling limits of discrete holomorphic functions up to
second order derivatives in the standard square lattices.
