MHB Apostol - Continuity & Differentiabilty

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The discussion centers on the proof of Apostol's Theorem 5.2, specifically regarding the continuity of the function $$f^*$$ at a point c. The user seeks a formal demonstration of this continuity, initially unsure about their approach. They propose that the limit $$\lim_{x \to c} f^* (x)$$ equals $$f'(c)$$, confirming that this aligns with the definition of continuity. Another participant validates this reasoning, providing reassurance. The conversation concludes with the user expressing gratitude for the confirmation of their understanding.
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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 $$f^*$$ 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
 
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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
 

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