Re: [分析] Global implicit function theorem
※ 引述《chopriabin ()》之銘言：
: Suppose that $f=f(x,y)$ is $C^1$ in $x$ and $y$,
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: 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 $x\in\mathbb{R}$?
Yes in a neighbourhood of x=x_0.
Globally, no, for example, f(x,y)=tanh(y)+x, x_0=y_0=0 and x>=1.
: 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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『我思故我在』怎樣從法文變成拉丁文的：
je pense, donc je suis --- René Descartes, Discours de la Méthode (1637)
ego sum, ego existo --- ____, Meditationes de Prima Philosophia (1641)
ego cogito, ergo sum --- ____, Principia Philosophiae (1644)
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