Let $A=\{(x,y)\in\mathbb{R}^2: x+y\neq -1\}$ and $f:A\to\mathbb{R}^2$, Define $f(x,y)=({x\over 1+x+y},{y\over 1+x+y})$,Then Check whether the function is infinite differentiable or not?
My attempt
method-1 If $f$ is infinitely differentiable iff Each component of $f$ is infinitely differentiable. Each component of $f$ is a rational function of two variable. denominator never vanishes. So, $f$ is one-time differentiable.
method- 2
$${h\over 1+h+k}-0=0.h+0.k+\sqrt{h^2+k^2}r(h,k)$$. We can easily prove that $\lim_{(h,k)\to(0,0)}r(h,k)=0$. Similarly for other component. Is there any short cut to check the infinite differentiability of given function?
method- 3 I tried to use the definition of the differentiability in the higher dimension. I couldn't find the corresponding linear map for each element in $A$. How do I found such linear transformation to prove the differentiability? If I define $T(x,y)=\frac{\partial(f_1,f_2)}{\partial(x,y)}(x,y)^t. $ Will it be fine? This is just the first derivative. How can I prove the existance of highere derivatives? Please help me.
A rational function with denominator nonzero on open set $A$ is $C^\infty$ on $A$. This follows by induction from the fact that it is once differentiable and the derivative is rational with denominator that is nonzero on $A$.