Continuous Functions on Intervals .... B&S Theorem 5.3.2 ....

  • #1
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I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ...

I am focused on Chapter 5: Continuous Functions ...

I need help in fully understanding an aspect of the proof of Theorem 5.3.2 ...Theorem 5.3.2 and its proof ... ... reads as follows:View attachment 7277In the above text from Bartle and Sherbert we read the following:

" Since is closed and the elements of belong to , it follows from Theorem 3.2.6 that . Then is continuous at ... ... "Can someone please explain exactly why/how we can conclude that is continuous at ... ?Peter
 
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  • #2
Peter said:
In the above text from Bartle and Sherbert we read the following:

" Since is closed and the elements of belong to , it follows from Theorem 3.2.6 that . Then is continuous at ... ... "Can someone please explain exactly why/how we can conclude that is continuous at ... ?
The theorem contains the hypothesis that is continuous on . By definition, that means that is continuous at each point of . The proof has already shown that . So it follows that is continuous at .
 
  • #3
Opalg said:
The theorem contains the hypothesis that is continuous on . By definition, that means that is continuous at each point of . The proof has already shown that . So it follows that is continuous at .
Oh ... careless of me not to notice that ...!

Thanks Opalg ...

Peter
 

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