Stick to two variables for convenience. Suppose we have some domain $D = \{(u, v) \in \mathbb{R}^2: a \leq u \leq b, c \leq v \leq d \}$, and we want to apply some change of variables function $x = f(u, v), y = g(u, v)$. We want a new domain (or rather a rephrasing of the old domain) $D'$ in terms of $x, y$. How we usually is by taking one leg each of the rectangle or triangle and finding the bounds for that particular line segment. We then stitch those line segments up to describe $D'$.
But doesn't this assume that the boundary of $D$ will become the boundary of $D'$ once we change the variables? How is this justified? Does it follow from continuity?