Exploring Theorem 4.29: Compact Metric Spaces & Inverse Functions

In summary, according to Theorem 4.16, if f^{-1} is continuous at a point, then it is continuous at all points in the domain of f^{-1}. However, in the example given, f^{-1} is not continuous at the point $f(0)$, because the sequence \{x_n\} does not converge in S.
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I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...

I am focused on Chapter 4: Limits and Continuity ... ...

I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as follows:

View attachment 9237
View attachment 9238
In the Example above we read the following:

" ... ... However, \(\displaystyle f^{ -1 }\) is not continuous at the point \(\displaystyle f(0)\). For example, if \(\displaystyle x_n = 1 - 1/n\), the sequence \(\displaystyle \{ f(x_n) \}\) converges to \(\displaystyle f(0)\) but \(\displaystyle \{ x_n \}\) does not converge in \(\displaystyle S\). ... ... "My question is as follows:

Can someone please explain exactly how/why ... the sequence \(\displaystyle \{ f(x_n) \}\) converges to \(\displaystyle f(0)\) but \(\displaystyle \{ x_n \}\) does not converge in \(\displaystyle S\) ... ... implies that \(\displaystyle f^{ -1 }\) is not continuous at the point \(\displaystyle f(0)\) ... ... ?

-----------------------------------------------------------------------------------------------------------------------------------------------My thoughts ...

I think that the relevant theorem regarding answering my question is Apostol, Theorem 4.16 which reads as follows:View attachment 9239

If we let \(\displaystyle t_n \in f(S)\) be such that \(\displaystyle t_n = f(x_n)\) ... so the sequence \(\displaystyle \{ t_n \} = \{ f(x_n) \}\) is in the domain of \(\displaystyle f^{ -1 }\) ...

Then ... sequence \(\displaystyle \{ t_n \} = \[ f(x_n) \}\) converges to \(\displaystyle f(0) = t_0\) say ...

Then following Theorem 4.16 above ... for \(\displaystyle f^{ -1 }\) to be continuous we need \(\displaystyle \{ f^{ -1 } (t_n) \} = \{ f^{ -1 } ( f(x_n) ) \} = \{ x_n \}\) to converge in \(\displaystyle S\) ... but it does not do so ...

(mind you ... I'm not sure how to prove it doesn't converge in \(\displaystyle S\) ...)

Is that correct?

-------------------------------------------------------------------------------------------------------------------------------------------------------------------

Hope that someone can help ...

Peter
 

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Hi Peter,

Your idea is correct. Now, the sequence $\{x_n\}$ converges to $1$ in $\mathbb{R}$, but, as $S$ does not contain $1$, that sequence does not converge in $S$.

Intuitively, what happens is that, when a point $P$ travels counterclockwise in circles in $f(S)$, the image $f^{-1}(P)$ jumps discontinuously from $(1-\varepsilon)$ to $0$ in $S$ whenever $P$ passes through $f(0)=1$.
 
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  • #3
castor28 said:
Hi Peter,

Your idea is correct. Now, the sequence $\{x_n\}$ converges to $1$ in $\mathbb{R}$, but, as $S$ does not contain $1$, that sequence does not converge in $S$.

Intuitively, what happens is that, when a point $P$ travels counterclockwise in circles in $f(S)$, the image $f^{-1}(P)$ jumps discontinuously from $(1-\varepsilon)$ to $0$ in $S$ whenever $P$ passes through $f(0)=1$.
Thanks castor28

Appreciate your help ...

Peter
 

1. What is Theorem 4.29 in compact metric spaces?

Theorem 4.29 states that if a function is continuous and bijective on a compact metric space, then its inverse function is also continuous.

2. What is a compact metric space?

A compact metric space is a metric space that is both complete and totally bounded. This means that every Cauchy sequence in the space converges to a point within the space, and the space can be covered by a finite number of open balls with a specific radius.

3. How does Theorem 4.29 relate to inverse functions?

Theorem 4.29 shows that the continuity of a function carries over to its inverse function on a compact metric space. This means that if a function is continuous and bijective on a compact metric space, then its inverse function will also be continuous.

4. Can Theorem 4.29 be applied to non-compact metric spaces?

No, Theorem 4.29 only applies to compact metric spaces. In non-compact metric spaces, the continuity of a function does not necessarily imply the continuity of its inverse.

5. How is Theorem 4.29 useful in mathematics and science?

Theorem 4.29 is useful in many areas of mathematics and science, including topology, analysis, and functional analysis. It allows for the study and manipulation of inverse functions in compact metric spaces, which has applications in fields such as physics, engineering, and computer science.

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