Consider the viscosity super solution to the equation $u_t - |u_x|=0$ given by $u(t,x) = \begin{cases} t+x \,\,\, \text{if } x \in (-t,0) \\ t-x \,\,\, \text{if } x \in (0,t) \\ 0 \,\,\, \text{else} \end{cases}$
I am having trouble seeing why this is a super-solution at the corners, and why it fails to be a sub solution at $x=0$. Could someone explain this to me?