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

In summary, the conversation discusses a theorem and its proof in "Introduction to Real Analysis" by Robert G Bartle and Donald R Sherbert. The theorem states that if $f$ is continuous on $I$, then $f$ is continuous at each point of $I$. The conversation focuses on understanding how this conclusion can be reached based on the given information.
  • #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 \(\displaystyle I\) is closed and the elements of \(\displaystyle X'\) belong to \(\displaystyle I\), it follows from Theorem 3.2.6 that \(\displaystyle x \in I\). Then \(\displaystyle f\) is continuous at \(\displaystyle x\) ... ... "Can someone please explain exactly why/how we can conclude that \(\displaystyle f\) is continuous at \(\displaystyle x\) ... ?Peter
 
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  • #2
Peter said:
In the above text from Bartle and Sherbert we read the following:

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

Thanks Opalg ...

Peter
 

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

What is the B&S Theorem 5.3.2?

The B&S Theorem 5.3.2, also known as the Bolzano-Weierstrass Theorem, states that every bounded sequence in the real numbers has a convergent subsequence.

What is the significance of the B&S Theorem 5.3.2?

The B&S Theorem 5.3.2 is a fundamental result in real analysis and has many applications in mathematics and other fields such as physics and engineering. It is also used to prove other important theorems, such as the Intermediate Value Theorem and the Extreme Value Theorem.

How does the B&S Theorem 5.3.2 relate to continuous functions on intervals?

The B&S Theorem 5.3.2 is often used to prove that continuous functions on intervals have certain properties, such as the Intermediate Value Property. This theorem guarantees the existence of a point where the function takes on every value between two given points.

Can the B&S Theorem 5.3.2 be applied to functions on any interval?

Yes, the B&S Theorem 5.3.2 can be applied to functions on any interval, as long as the function is continuous on that interval. This theorem is a general result that applies to any bounded sequence in the real numbers.

Are there any limitations to the B&S Theorem 5.3.2?

The B&S Theorem 5.3.2 only applies to functions on intervals in the real numbers. It does not apply to functions on other number systems, such as complex numbers. Additionally, it requires the function to be continuous on the interval, so it cannot be applied to discontinuous functions.

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