What would the parallellepiped of continuity look like?

Question

I'm going through Hartman's ODE, and I'm having trouble coming up with some examples where the region of continuity is not a cube looks like. For reference, he uses the definition $$R:t_0<t<t_0+α,|y−y_0|<b$$ where $y_0$ is the initial condition a lot, but I can't see any case where this would not be some n-cube. One potential case I can think of is if $y^1,y^2$ are the characteristics of the wave equation, but everytime I think about it it seems like I come to the opposite conclusion. Are there any "standard" examples, or does it simply depend on the space of the solution $y$?