Understanding Browder Proposition 3.14: Increasing Function & Discontinuities

In summary, Browder Proposition 3.14 is a mathematical concept that deals with increasing functions and discontinuities. It states that an increasing function must have a finite number of discontinuities. This concept helps to explain the behavior of increasing functions and is important for analyzing and graphing them accurately. It also serves as a tool for identifying errors in mathematical models involving increasing functions.
  • #1
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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:
View attachment 9536
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:
View attachment 9537
Hope that helps ...

Peter
 

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  • #2
Re: Increasing Function and Discontinuitiesl ... Browder, Proposition 3.14 ... ...

Peter said:
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
 
  • #3

Dear Peter,

Thank you for your question. I can understand your confusion with Proposition 3.14 and its proof. Let me try to explain it in a more detailed manner.

Proposition 3.7 states that for any point d in the interval I, which is not an endpoint of I, the left and right limits of a function f exist and are bounded by the value of f at d. In other words, we can say that the function f is continuous at d.

Now, in Proposition 3.14, we are looking at a specific set D, which is defined as the set of points in the interval I where the left and right limits of f are not equal. In other words, at these points, the function is not continuous.

In the proof of Proposition 3.14, Browder is showing that if d is in D, then the left and right limits of f at d are not equal, which is the definition of D. On the other hand, if d is not in D, then the left and right limits of f at d are equal, and therefore the function is continuous at d.

Now, to answer your question, we need to understand why d is in D if and only if f(d-) < f(d+). This is because if d is in D, it means that the left and right limits of f at d are not equal, and therefore one is strictly less than the other. On the other hand, if f(d-) < f(d+), it means that the left and right limits of f at d are not equal, and therefore d is in D.

I hope this explanation helps you understand Proposition 3.14 better. If you have any further questions, please feel free to ask.
 

FAQ: Understanding Browder Proposition 3.14: Increasing Function & Discontinuities

What is Browder Proposition 3.14?

Browder Proposition 3.14 is a mathematical theorem that states if a function is increasing on an interval, then it is continuous on that interval.

What does it mean for a function to be increasing?

A function is considered increasing if its values increase as the input values increase. In other words, as the x-values increase, the corresponding y-values also increase.

What are discontinuities in a function?

Discontinuities in a function occur when there is a break or gap in the graph of the function. This means that the function is not continuous at that point.

How does Browder Proposition 3.14 relate to calculus?

Browder Proposition 3.14 is a fundamental theorem in calculus that helps us understand the relationship between increasing functions and continuity. It is often used to prove other theorems and solve problems in calculus.

Can Browder Proposition 3.14 be applied to all functions?

Yes, Browder Proposition 3.14 can be applied to all functions that are increasing on an interval. However, it is important to note that not all functions are increasing, and therefore this theorem may not be applicable in all cases.

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