Functions Continuous on Comapct Sets .... Apostol, Theorem 4.25 ....

In summary, the conversation discusses the proof of Theorem 4.25 from Tom M Apostol's book "Mathematical Analysis" (Second Edition) and requests for an explicit formal and rigorous proof of the statement that the sets f^{ -1 } (A) form an open covering of X. The conversation also includes a reference to Apostol Exercise 2.7 and a proposed proof by another member. The final response provides a revised and convincing proof.
  • #1
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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 proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows:
View attachment 9231
View attachment 9232

In the above proof by Apostol we read the following:

" ... ... The sets \(\displaystyle f^{ -1 } (A)\) form an open covering of \(\displaystyle X\) ... ... "Could someone please demonstrate an explicit formal and rigorous proof of this statement ...?

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My thoughts:

Since \(\displaystyle f\) is continuous we have that each set \(\displaystyle f^{ -1 } (A)\) is open

and ...

... for \(\displaystyle X \subseteq S\) we have

\(\displaystyle X \subseteq f^{ -1 } ( f(x) )\) ... ... (see Apostol Exercise 2.7 (a) Chapter 2, page 44 ...)

... and we also have \(\displaystyle f(X) \subseteq A_c\) where \(\displaystyle A_c = \bigcup_{ A \in F } A\) ...

Therefore \(\displaystyle X \subseteq f^{ -1 } ( f(x) ) \subseteq f^{ -1 } ( A_c )\) ...Is that correct? ... Does that constitute a formal and rigorous proof?

Hope someone can help ...

Peter

================================================================================The above post refers to Apostol Exercise 2.7 so I am providing access to the same as follows:
View attachment 9233
Hope that helps ...

Peter
 

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  • #2
Peter said:
In the above proof by Apostol we read the following:

" ... ... The sets \(\displaystyle f^{ -1 } (A)\) form an open covering of \(\displaystyle X\) ... ... "Could someone please demonstrate an explicit formal and rigorous proof of this statement ...?
I would put it like this:

If $x\in X$ then $f(x) \in f(X) \subseteq \bigcup_{A\in F}A$. Therefore $f(x)$ is in (at least) one of the sets $A \in F$, say $f(x)\in A_x$. Then $x \in f^{-1}(A_x)$. Since that holds for every $x$ in $X$, it follows that the sets \(\displaystyle f^{ -1 } (A)\) form a covering of \(\displaystyle X\).
 
  • #3
Opalg said:
I would put it like this:

If $x\in X$ then $f(x) \in f(X) \subseteq \bigcup_{A\in F}A$. Therefore $f(x)$ is in (at least) one of the sets $A \in F$, say $f(x)\in A_x$. Then $x \in f^{-1}(A_x)$. Since that holds for every $x$ in $X$, it follows that the sets \(\displaystyle f^{ -1 } (A)\) form a covering of \(\displaystyle X\).
Thanks for a very convincing proof Opalg ...

Appreciate your help ...

Peter
 

1. What is the significance of a function being continuous on a compact set?

A function being continuous on a compact set means that the function is defined and has no interruptions on a closed and bounded interval. This is important because it allows us to make predictions and analyze the behavior of the function over the entire interval, rather than just at specific points.

2. How do you determine if a function is continuous on a compact set?

To determine if a function is continuous on a compact set, we need to check that the function is defined on the entire interval and that the limit of the function exists at every point within the interval. If both of these conditions are met, then the function is continuous on the compact set.

3. What is the Apostol Theorem 4.25?

The Apostol Theorem 4.25 states that if a function is continuous on a compact set, then the function is also uniformly continuous on that set. This means that the function's behavior is consistent and predictable over the entire interval, rather than just at specific points.

4. How can we apply the Apostol Theorem 4.25 to real-world situations?

The Apostol Theorem 4.25 can be applied to real-world situations in various fields such as physics, engineering, and economics. For example, in physics, we can use this theorem to analyze the behavior of a continuous function representing the motion of an object over a specific time interval. In economics, we can use it to study the demand and supply of a product over a given time period.

5. What are some examples of functions that are continuous on compact sets?

Examples of functions that are continuous on compact sets include polynomials, trigonometric functions, and exponential functions. For instance, the function f(x) = x^2 is continuous on the compact set [0,1], as it is defined and has no interruptions on this interval. Similarly, the function g(x) = sin(x) is continuous on the compact set [0,2π], as it is defined and has no interruptions on this interval.

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