In the principle of maximum for solutions of the heat equation, the hypothesis is that I have a function $u: \bar \Omega\times [0,T] \to R$ with $\Omega \in \mathbb{R}^n$ and $u \in C(\bar{\Omega} \times [0,T])$, $\nabla_{x,t} u \in C(\bar \Omega \times \{t\}) \forall t \in [0,T]$, $\partial_{x_i} \partial_{x_j} u \in C(\bar \Omega \times [0,T])$ and $\partial_t u - \Delta_x u=0$
If I am interpreting the second condition correctly, it is saying that at fixed time, the function is continuous in space, but it is not in time. However the solution of the heat equation with a continuous and limited function on the border should be unique and well-behaved with respect to the initial condition.
The question is: what is an example of a function that satisfies these hypothesis, but whose gradient is not continuous on $\Omega \times [0,T]$? If I am not mistaken, this should be equivalent to asking for a function satisfying the heat equation $\partial_t u - \Delta_x u=0$ but that does not depend continuously by the initial condition.