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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.enter image description here

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.

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    @Vikram: No, that is not the reason your set-up is confusing. Your set-up is confusing because you **explicitly** state that you are considering a domain that does not include $(0,0)$, and yet you also claim to be trying to find the limit as $(x,y)$ approaches $(0,0)$. You are not just confused about what you are trying to do, you are confused about what it is you are confused about. You **explicitly** claim in your post that you are considering as domain **only** the **blue region**. The blue region **does not** contain $(0,0)$. So it's not a problem of 2d vs 3d.2012-04-14

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In 1-D the epsilon-delta definition of continuity at a point $x_0$ is $\forall \epsilon>0$, $\exists \delta>0$ such that $|f(x_0+\epsilon)-f(x_0)|<\delta$.

You said

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

but actually the way to extend the idea is to consider a ϵ-width ball around $\vec{x_0}$. Then the epsilon-delta definition will be $\forall \vec{\epsilon}$ with $|\vec{\epsilon}|>0$, $\exists \delta>0$ such that $\|f(\vec{x_0}+\vec{\epsilon})-f(\vec{x_0})\|<\delta$.

In English, the continuity holds in every direction around the point $\vec{x_0}$.