Articles of inverse function theorem

An application of the Inverse function theorem

I have recently come across two formula’s that I am unfamiliar with and would like to know if they are both aspects of the same thing: $$\color{purple}{\cfrac{1}{f^{\prime}(a)}=f(a)(f^{-1})^{\prime}}\tag{1}$$ $$\rho_x (x)=\rho_\alpha(\alpha)\left|\frac{\mathrm{d}x}{\mathrm{d}\alpha}\right|^{-1}\tag{2}$$ $(1)$ is the Inverse function theorem and $(\mathrm{2})$ is the formula that relates probability densities for random variables $x$, $\alpha$. In my previous post I was […]

Why do we want TWO open sets from the inverse function theorem?

I have been analyzing Rudin’s proof of the Inverse Function Theorem closely over the last two days, and trying to understand what the purpose of every assumption made is. The first assumption that he makes is the value of the radius $\lambda$ of the neighborhood U such that $2\lambda = \frac{1}{\lVert f'(a)^{-1} \rVert}$. This definition […]

How is the Inverse function theorem used to prove that the formulae in this question are the same?

I was informed in my last question that the Inverse function theorem: $$(f^{-1})^{\prime}(f(a))=\cfrac{1}{f^{\prime}(a)}\tag{I*}$$ was needed to show that $$\rho_x (x)=\rho_\alpha(\alpha)\left|\frac{\mathrm{d}x}{\mathrm{d}\alpha}\right|^{-1}\tag{A}$$ is the same formula as $$\rho_y(y)=\rho_x(\phi^{-1}(y))\left|\frac{\mathrm{d}\phi^{-1}}{\mathrm{d}y}\right|\tag{B}$$ Where $(\mathrm{A})$ and $(\mathrm{B})$ are formulae that relate the probability densities for random variables $x$, $\alpha$ in $(\mathrm{A})$ and $y$ and $x$ in $(\mathrm{B})$. It’s probably pretty simple to […]