We give conditions under which bounded solutions to semilinear elliptic equations Δ u = f ( u ) \Delta u = f(u) on domains of R 2 \mathbb {R}^2 are continuous despite a possible infinite singularity of f ( u ) f(u) . The conditions do not require a minimization or variational stability property for the solutions. The results are used in a second paper to show regularity for a familiar class of equations.