  We establish the local continuity of locally bounded weak solutions
(temperatures) to the doubly singular parabolic equation modeling the phase
transition of a material: \[ \partial_t \beta(u)-\Delta_p u\ni 0\quad\text{ for
}\tfrac{2N}{N+1}<p<2, \] where $\beta$ is a maximal monotone graph with a jump
at zero and $\Delta_p$ is the $p$-Laplacian. Moreover, a logarithmic type
modulus of continuity is quantified, which has been conjectured to be optimal.
