When we check for the continuity of a function of one variable, we check the left hand side and right hand side limit of the function about the point in question, clearly this point lies inside the domain, what I mean is that if we are checking the continuity at $c$ then $a\leq c\leq b$, extending the same idea to functions with two variables, when we check the function's behavior/continuity at some point, we check it along two different paths and I have seen some examples where the author considers X-axis, Y-axis, $y=x$, $y=kx^2$ etc, my question is, these paths have to be included in the region under investigation, what I understand is that if we have a function of two variables, first we will have to draw the region defined by the function for various values of the independent variables $x$ and $y$ and then choose two different paths included in that region.
As shown in the above diagram, if I get the region as shown by the Blue area enclosed by the line $y=x$ and $y=0$ from $x=2$ to $x \to \infty$, the curve $y=x^2$ is not included in the region so I can not use it to test the continuity, but in the example that I have seen author does not mention that the path he is choosing lies inside the region.