  We address the problem of an unscreened Coulomb charge in graphene, and
calculate the local density of states and displaced charge as a function of
energy and distance from the impurity. This is done non-perturbatively in two
different ways: (1) solving the problem exactly by studying numerically the
tight-binding model on the lattice; (2) using the continuum description in
terms of the 2D Dirac equation. We show that the Dirac equation, when properly
regularized, provides a qualitative and quantitative low energy description of
the problem. The lattice solution shows extra features that cannot be described
by the Dirac equation, namely bound state formation and strong renormalization
of the van Hove singularities.
