Working on an old exam question:
Show that for every $f \in C(\mathbb{T}), \epsilon > 0$, there is an initial condition $g \in C(\mathbb{T})$ for which there is a solution $u(x,t)$ to the heat equation on a ring with $u(x,0) = g(x)$ and $|u(x,1) - f(x)| < \epsilon$ for every $x \in \mathbb{T}$.
I think it wants me to show that given a certain boundary condition $f(x)$, there must exist some initial condition $g(x)$ for which the heat equation has a solution. But I'm not sure how to approach this proof.