We show continuity of bounded positive solutions to some semilinear elliptic equations $\Delta u =f(u)$ in domains of $\mathbb R^2$. Our technique involves ruling out sequences of solutions that we call ``tornado solutions,'' and does not assume the variational stability that is required in other treatments of these equations. We will also discuss the use of our results in constructing singular solutions, alternative techniques of other authors, and the relation to the general study of singularities of minimal hypersurfaces in Euclidean space.