[分析] Global implicit function theorem
Suppose that $f=f(x,y)$ is $C^1$ in $x$ and $y$,
where $x,y\in\mathbb{R}$ and $f(x_0,y_0)=0$.
Moreover, $\frac{\partial f(x,y)}{\partial y}>0$ for all $x,y\in\mathbb{R}$.
Could we conclude that $y$ can be solved implicitly from $f(x,y)=0$
in terms of $x$ as $y=y(x)$ for all $x\in\mathbb{R}$?
If the answer is no and an additional assumption is imposed,
i.e. we can obtain $y'(x)>0$ for all $x\in\mathbb{R}$
by implicitly differentiating $f(x,y(x))=0$ with respect to $x$,
then these assumptions are enough to get the same conclusion?
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