We study nonlinear potential theory on a metric measure
space equipped with a doubling measure and supporting a
Poincaré inequality. Minimizers, superminimizers and the
obstacle problem for the $p$-Dirichlet integral play an
important role in the theory. We prove lower semicontinuity of
superminimizers and continuity of the solution to the obstacle
problem with a continuous obstacle. We also show that the
limit of an increasing sequence of superminimizers is a
superminimizer provided it is bounded above. Moreover, we
consider superharmonic functions and study their relations to
superminimizers. Our proofs are based on the direct methods of
the calculus of variations and on De Giorgi type estimates. In
particular, we do not use the Euler-Lagrange equation and our
arguments are based on the variational integral only.