  By a symmetric double graph we mean a hypersurface which is mirror-symmetric
and the two symmetric parts are graphs over the hyperplane of symmetry. We
prove that there is a weak solution of mean curvature flow that preserves these
properties and singularities only occur on the hyperplane of symmetry. The
result can be used to construct smooth solutions to the free Neumann boundary
problem on a supporting hyperplane with singular boundary. For the construction
we introduce and investigate a notion named "vanity" and which is similar to
convexity. Moreover, we rely on S\'aez' and Schn\"urer's "mean curvature flow
without singularities" to approximate weak solutions with smooth graphical
solutions in one dimension higher.
