  We study viscosity solutions to the classical one-phase problem and its thin
counterpart. In low dimensions, we show that when the free boundary is the
graph of a continuous function, the solution is the half-plane solution. This
answers, in the salient dimensions, a one-phase free boundary analogue of
Bernstein's problem for minimal surfaces.
  As an application, we also classify monotone solutions of semilinear
equations with a bump-type nonlinearity.
