Apostol - Continuity & Differentiabilty

  • Context: MHB 
  • Thread starter Thread starter Math Amateur
  • Start date Start date
  • Tags Tags
    Apostol Continuity
Click For Summary
SUMMARY

The discussion centers on Apostol's Theorem 5.2, specifically the proof of the continuity of the function $$f^*$$ at a point c. The user Peter seeks confirmation on his formulation of the limit, which states that $$\lim_{x \to c} f^* (x) = \lim_{x \to c} \frac{f(x) - f(c)}{x-c} = f'(c) = f^*(c)$$. Other participants affirm that this formulation aligns with the definition of continuity, providing Peter with the confidence he needed regarding his understanding of the theorem.

PREREQUISITES
  • Understanding of Apostol's Theorem 5.2
  • Knowledge of limits and continuity in calculus
  • Familiarity with the concept of derivatives
  • Basic proficiency in mathematical proofs
NEXT STEPS
  • Study the formal proof of Apostol's Theorem 5.2 in detail
  • Explore the definition of continuity in mathematical analysis
  • Learn about the properties of derivatives and their implications
  • Review examples of limit proofs in calculus
USEFUL FOR

Students and educators in mathematics, particularly those focusing on calculus and real analysis, as well as anyone interested in understanding the rigorous foundations of continuity and differentiability.

Math Amateur
Gold Member
MHB
Messages
3,920
Reaction score
48
I need help with the proof of Apostol Theorem 5.2.

Theorem 5.2 and its proof read as follows:
https://www.physicsforums.com/attachments/3910
In the above proof, Apostol gives an expression or formula for $$f^*$$ and then states the following:

" ... ... Then $$f^*$$ is continuous at c ... ... "I need help with formulating a rigorous and formal demonstration that $$f^*$$ is continuous ...

Can someone please help?

Peter***EDIT***

Oh! Just had a thought regarding my question above ... ...

Presumably the demonstration I was looking for is simply the following ... ...$$ \lim_{x \to c} f^* (x) = \lim_{x \to c} \frac{f(x) - f(c)}{x-c} = f'(c) = f^*(c) $$

Can someone please confirm that this is correct?

Peter
 
Last edited:
Physics news on Phys.org
That is exactly the definition of continuity so imo that is correct.
 
Siron said:
That is exactly the definition of continuity so imo that is correct.

Thanks Siron ... just needed some confidence that I was correct ... so thanks!

Peter
 

Similar threads

MHB Functions Continuous on Comapct Sets .... Apostol, Theorem 4.25 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Complex Derivative .... Remark in Apostol, Section 16.1 .... ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Directional Derivatives .... Apostol, Section 12.2, Example 4 ....
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....
  • · Replies 2 ·
Replies
2
Views
4K
MHB How Does Theorem 4.29 Illustrate Continuity Issues in Inverse Functions?
  • · Replies 2 ·
Replies
2
Views
2K
MHB Proof of Apostol's Definition 3.2 and Theorem 3.3: Help Appreciated
  • · Replies 4 ·
Replies
4
Views
2K
MHB Lindelof Covering Theorem .... Apostol, Theorem 3.28 ....
  • · Replies 2 ·
Replies
2
Views
2K
MHB Compactness and Uniform Continuity in R^n .... .... D&K Theorem 1.8.15
  • · Replies 2 ·
Replies
2
Views
2K
MHB Continuity and Compact Sets - Bolzano's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
MHB Metric Spaces & Compactness - Apostol Theorem 4.28
  • · Replies 3 ·
Replies
3
Views
2K