We establish the continuity of bounded local solutions of the equation $\beta {\left(u\right)}_t=\mathrm{\Delta }u$. Here $\beta$ is any coercive maximal monotone graph in $\mathbb{R}\times \mathbb{R}$, bounded for bounded values of its argument. The multiphase Stefan problem and the Buckley-Leverett model of two immiscible fluids in a porous medium give rise to such singular equations.