I have an optimization problem whose value function I denote by $U(x,t)$ where $x\in [0,1]$ is a state variable and $t\in [0,1]$ is time. The HJB equation for my optimization problem when $x<1$ is given by
$$ U(x,t)=\frac{\partial U(x,t)}{\partial t}-\frac{\partial^2 U(x,t)}{\partial x^2}(1-x) $$ with boundary condition $U(x,1)=Y(x)$, where $Y(x)$ is a continuous function with $\lim_{x\to 1} Y(x)=y$.
The HJB equation for $x=1$ is instead given by $$ U(1,t)=\frac{\partial U(1,t)}{\partial t}-C $$ where $C>0$ is a constant, with boundary conditions $U(1,1)=y$.
Does this suggest that $U(x,t)$ is not continuous at $x=1$ when $t<1$. If so, does that mean the solution I derive from the HJB equation is not (necessarily) the solution to the optimization problem?