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Increasing Function and Discontinuities ... Browder, Proposition 3.14 ... ...

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.14 ...


Proposition 3.14 and its proof read as follows:



Browder - Proposition 3.14  .png



In the above proof by Browder we read the following:

" ... ... For any \(\displaystyle d \in I, d\) not an endpoint of \(\displaystyle I\) we know (Proposition 3.7) that \(\displaystyle f(d-)\) and \(\displaystyle f(d+)\) exist with \(\displaystyle f(d-) \leq f(d) \leq f(d+)\), so \(\displaystyle d \in D\) if and only if \(\displaystyle f(d-) \lt f(d+)\). ... ... "


My question is as follows:

Can someone please demonstrate (rigorously) exactly why/how it follows that \(\displaystyle d \in D\) if and only if \(\displaystyle f(d-) \lt f(d+)\). ... ...


Help will be much appreciated ...

Peter



=======================================================================================


The above post mentions Browder Proposition 3.7 ... so I am providing the text of that proposition ... as follows:



Browder - Proposition 3.7 ... .png



Hope that helps ...

Peter
 
Last edited:

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,891
Re: Increasing Function and Discontinuitiesl ... Browder, Proposition 3.14 ... ...

I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ...

I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ...

I need some help in understanding the proof of Proposition 3.14 ...


Proposition 3.14 and its proof read as follows:







In the above proof by Browder we read the following:

" ... ... For any \(\displaystyle d \in I, d\) not an endpoint of \(\displaystyle I\) we know (Proposition 3.7) that \(\displaystyle f(d-)\) and \(\displaystyle f(d+)\) exist with \(\displaystyle f(d-) \leq f(d) \leq f(d+)\), so \(\displaystyle d \in D\) if and only if \(\displaystyle f(d-) \lt f(d+)\). ... ... "


My question is as follows:

Can someone please demonstrate (rigorously) exactly why/how it follows that \(\displaystyle d \in D\) if and only if \(\displaystyle f(d-) \lt f(d+)\). ... ...


Help will be much appreciated ...

Peter



=======================================================================================


The above post mentions Browder Proposition 3.7 ... so I am providing the text of that proposition ... as follows:







Hope that helps ...

Peter




After a little reflection I have realized that \(\displaystyle f\) is continuous at any point \(\displaystyle x\) if and only if \(\displaystyle f(x-) = f(x+)\) ... so if f is discontinuous at \(\displaystyle d\) then \(\displaystyle f(d-) \neq f(d+)\) ... but ... we have that \(\displaystyle f(d-) \leq f(d+)\) ... so therefore \(\displaystyle f(d-) \lt f(d+)\) ...


Is that correct?

Peter