  Extending DiNezza-Lu's approach to the setting of big cohomology classes, we
prove that solutions of degenerate complex Monge-Amp{\`e}re equations on
compact K{\"a}hler manifolds are continuous on a Zariski open set. This allows
us to show that singular K{\"a}hler-Einstein metrics on log canonical varieties
of general type have continuous potentials on the ample locus outside of the
non-klt part.
