To JohnAndrews comments, you might want to add
As Area goes to zero, Stress goes to infinity (this is sometimes called the Boussinesq problem).
Further, as the stress goes to infinity in this case, the strain energy goes to infinity.
This is a very important result that is almost never discussed in finite element classes or outside of the university. This result addresses the convergence of the numerical solution. It can be shown (Szabo and Babuska, Finite Element Analysis), that the minimization of the potential energy (of which strain energy is a part) is equivalent to finding the exact solution. If you have a situation in which the strain energy (and, potential energy) is infinite, your finite element solution cannot be fully trusted because obtaining a converged finite element solution depends on minimization of the potential energy--you are searching for the exact solution with your finite element analysis; if you cannot minimize the potential energy because the exact solution's strain energy is infinite, then you cannot obtain a converged finite element solution. This is why a so-called "Point Load" or "Point Displacement" constraint is not allowed with the finite element method.
While it appears that this Boussinesq problem is a singularity similar to the singularity of a crack, the key difference is the behavior of the strain energy in the Boussinesq problem, which is infinite, compared to the crack problem, which has finite or bounded strain energy.