Apostol - Continuity & Differentiabilty

In summary, the conversation is about the proof of Apostol Theorem 5.2 and how to demonstrate the continuity of f^*. The person, Peter, asks for help in formulating a rigorous and formal demonstration, but then realizes that the demonstration is simply the definition of continuity. The conversation ends with Siron confirming that Peter's understanding is correct.
  • #1
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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 \(\displaystyle f^*\) and then states the following:

" ... ... Then \(\displaystyle f^*\) is continuous at c ... ... "I need help with formulating a rigorous and formal demonstration that \(\displaystyle f^*\) 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 ... ...\(\displaystyle \lim_{x \to c} f^* (x) = \lim_{x \to c} \frac{f(x) - f(c)}{x-c} = f'(c) = f^*(c) \)

Can someone please confirm that this is correct?

Peter
 
Last edited:
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  • #2
That is exactly the definition of continuity so imo that is correct.
 
  • #3
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
 

1) What is the concept of continuity in Apostol - Continuity & Differentiability?

Continuity is the property of a function where there are no abrupt changes or breaks in the graph. In other words, a function is continuous if the limit of the function at a given point is equal to the value of the function at that point.

2) How is continuity different from differentiability?

Continuity and differentiability are related concepts but they are not the same. A function is continuous if it has no abrupt changes, while a function is differentiable if it has a well-defined derivative at every point. In other words, a function can be continuous but not differentiable, but if a function is differentiable, then it must also be continuous.

3) What is the difference between a removable and non-removable discontinuity?

A removable discontinuity is a point where a function is not continuous but can be made continuous by redefining the value of the function at that point. On the other hand, a non-removable discontinuity is a point where a function is not continuous and cannot be made continuous by redefining the value of the function.

4) How can I determine if a function is continuous or differentiable?

To determine if a function is continuous, you can use the definition of continuity and check if the limit of the function at a given point is equal to the value of the function at that point. To determine if a function is differentiable, you can use the definition of differentiability and check if the derivative of the function exists at every point.

5) What are some real-life applications of continuity and differentiability?

Continuity and differentiability are essential concepts in calculus and are used in various fields such as physics, engineering, economics, and more. Some real-life applications include calculating rates of change, determining maximum and minimum values, and analyzing the behavior of functions in real-world situations.

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