Inverse Function Theorem for One Real Variable

[Math Amateur]

I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of the Inverse Function Theorem (IFT) for real-valued functions of one real variable.

Stoll's statement of the IFT for Derivatives and its proof read as follows:
View attachment 3934

In the above proof we read:

" ... ... Since \(\displaystyle f^{-1}\) is continuous, \(\displaystyle x_n \rightarrow x_0 = f^{-1} (y_0)\) ... ... "I do not understand what Stoll means by this statement ... indeed it may be a 'typo' ... and if it is, what did he mean to say ...

Worse still for my understanding of this proof I cannot see where the continuity of \(\displaystyle f^{-1}\) is required in the proof of:

\(\displaystyle ( f^{-1} )' (y_0) = \frac{1}{f' (x_0 )} \)Can someone please help clarify the above situation?

Peter

 

[Fallen Angel]

Hi Peter,

Continuity is needed, it's not a typo.

He tries to prove that $f^{-1}(y_0)=x_0$

For any real valued continuous (at $a$) function $h$ and any sequence $\{a_n \}_{n\in \mathbb{N}}$ with limit $a$ we got that $\underset{n \to \infty}\lim h(a_{n})=h(\underset{n \to \infty}\lim a_{n})=h(a)$, which is not true (in general) for non continuous functions (at $a$).

So he needs continuity to guarantee that $\underset{n\to \infty}\lim f^{-1}(y_{n})=f^{-1}(y_{0})$

 

[Math Amateur]

I am reading Manfred Stoll's book: Introduction to Real Analysis.

I need help with Stoll's proof of the Inverse Function Theorem (IFT) for real-valued functions of one real variable.

Stoll's statement of the IFT for Derivatives and its proof read as follows:In the above proof we read:

" ... ... Since \(\displaystyle f^{-1}\) is continuous, \(\displaystyle x_n \rightarrow x_0 = f^{-1} (y_0)\) ... ... "I do not understand what Stoll means by this statement ... indeed it may be a 'typo' ... and if it is, what did he mean to say ...

Worse still for my understanding of this proof I cannot see where the continuity of \(\displaystyle f^{-1}\) is required in the proof of:

\(\displaystyle ( f^{-1} )' (y_0) = \frac{1}{f' (x_0 )} \)Can someone please help clarify the above situation?

Peter

Hi Peter,

Continuity is needed, it's not a typo.

He tries to prove that $f^{-1}(y_0)=x_0$

For any real valued continuous (at $a$) function $h$ and any sequence $\{a_n \}_{n\in \mathbb{N}}$ with limit $a$ we got that $\underset{n \to \infty}\lim h(a_{n})=h(\underset{n \to \infty}\lim a_{n})=h(a)$, which is not true (in general) for non continuous functions (at $a$).

So he needs continuity to guarantee that $\underset{n\to \infty}\lim f^{-1}(y_{n})=f^{-1}(y_{0})$

Thanks for the help, Fallen Angel ... appreciate your support ...

Peter