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$X:= f^{-1} (\{0\})=\{(x,y) \in \mathbb{R^2} | f(x,y)=0\}$

1. Show that $f$ is continuous differentiable.

2. For which $(x,y) \in \mathbb{R^2}$ is the implicit function theorem usable to express $y$ under the condition $f(x,y)=0$ as a function of $x$?

3. Let $(a,b) \in X$ with $b>0$. Find the largest possible neighbourhood $V$ of $a$ in $\mathbb{R}$ and a continuous differentiable function $g:V \rightarrow \mathbb{R}$ such that $f(x,g(x))=0$ and $g(a)=b$.

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1. $[df(x,y)]=(\frac{\delta f}{\delta x}, \frac{\delta f}{\delta y})=(2x, 2y)$

2. $0=x^2+y^2-1$ $\Rightarrow$ $y=\sqrt{-x^2+1}$, it follows that $x \in [-1,1]$ and $y \in [0,1]$.

Is 1 and 2 correct? How do I do 3?