$f_{r}(x,y;\xi) = f_{X} (x,y;\xi) $ when $y \leq g(x;\xi)$ and $f_{r}(x,y;\xi) = f_{Y} (x,y;\xi)$ when $y \geq g(x;\xi)$.
where $g(x;\xi) = 1 - p_{1}x - p_{3}\xi + O(x^2 + \xi^2)$ and $\xi$ is a real parameter.
$f_{r},g$ are $C^2$ functions.
To study $f_{X}$ and $f_{Y}$ near $(x,y) = (0,1)$ and for small values of $\xi$, we expand $f_{X}$ and $f_{Y}$ about $(x,y;\xi) = (0,1;0)$
Now since $f_{r}$ is continuous we have $f_{X} = f_{Y}$ on $y = g(x;\xi)$.
Now I am thinking about the next statement -
There exists constants $a_{1},a_{2},b_{1}^{X},b_{2}^{X},b_{1}^{Y},b_{2}^{Y},c_{1},c_{2} \in \Bbb{R}$ such that
$f_{S}(x,y;\xi) = \begin{cases} 1 + (a_{1}+b_{1}^{S}p_{1})x + b_{1}^{S}(y-1)+(c_{1}+b_{1}^{S}p_{3})\xi+O(x^2+(y-1)^2+\xi^2)\\ (a_{2}+b_{2}^Sp_{1})x+b_{2}^{S}(y-1)+(c_{2}+b_{2}^{S}p_{3})\xi +O(x^2+(y-1)^2+\xi^2) \end{cases} $
$S =X,Y$
I think we are using Taylor series, If so how are we using it?. Is this correct? any help?