  We employ a variational approach to study the Neumann boundary value problem
for the $p$-Laplacian on bounded smooth-enough domains in the metric setting,
and show that solutions exist and are bounded. The boundary data considered are
Borel measurable bounded functions. We also study boundary continuity
properties of the solutions. One of the key tools utilized is the trace theorem
for Newton-Sobolev functions, and another is an analog of the De Giorgi
inequality adapted to the Neumann problem.
