We have the PDE $u_t = u_{xx}$ with initial conditions $u(x, 0) = u_0(x)$ given. How does one show that $u(x, t) \leq \mathrm{max}_x\;u_0(x)$ for all $x$ and $t$?
I later have to show that a maximum principle also holds for a discretization of a PDE, so was curious to know how one goes about doing this for the continuous case?