Intermediate Value Theorem .... Browder, Theorem 3.16 .... ....

In summary, the proof of Theorem 3.16 in Browder's book involves demonstrating that the continuous image of a connected set is connected, and that an interval is connected if and only if it has a supremum in the closed interval [a,b].
  • #1
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I need help with an aspect of the proof of the Intermediate Value Theorem ...
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 Theorem 3.16 ...Theorem 3.16 and its proof read as follows:
Browder - 1 -  Theorem 3.16 ... ... PART 1 ... .png

Browder - 2 -  Theorem 3.16 ... ... PART 2 .png

In the above proof by Andrew Browder we read the following:

" ... ... But ##f(b) \gt y## implies (since ##f## is continuous at ##b##) that there exists ##\delta \gt 0## such that ##f(t) \gt y## for all ##t## with ##b - \delta \lt t \leq b##. ... ... "My question is as follows:

How do we demonstrate explicitly and rigorously that since ##f## is continuous at ##b## and ##f(b) \gt y## therefore we have that there exists ##\delta \gt 0## such that ##f(t) \gt y## for all ##t## with ##b - \delta \lt t \leq b##. ... ...Help will be much appreciated ...

Peter

***NOTE***

The relevant definition of one-sided continuity for the above is as follows:

##f## is continuous from the left at ##b## implies that for every ##\epsilon \gt 0## there exists ##\delta \gt 0## such that for all ##x \in [a, b]## ...

we have that ##b - \delta \lt x \lt b \Longrightarrow \mid f(x) - f(b) \mid \lt \epsilon##
 
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  • #2
I hate proofs like these. They obscure what's really happening. General topology and the notions of connectedness and compactness immediately imply your theorem.

However, the question you ask is very straightforward to anwer.

Since ##f(b)>y##, we have ##\epsilon:= f(b)-y>0##. Now use the definition of continuity with this choice of ##\epsilon##.

Draw a picture to see how I came up with this choice of ##\epsilon##.
 
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  • #3
Math_QED said:
I hate proofs like these. They obscure what's really happening. General topology and the notions of connectedness and compactness immediately imply your theorem.

However, the question you ask is very straightforward to anwer.

Since ##f(b)>y##, we have ##\epsilon:= f(b)-y>0##. Now use the definition of continuity with this choice of ##\epsilon##.

Draw a picture to see how I came up with this choice of ##\epsilon##.
Thank you for a most interesting and informative reply ...

Will now look at other proofs to see how concepts of compactness and connectedness imply theorem ...

Will also work on your suggestion ...

Thanks again,

Peter
 
  • #4
It is of course easy to prove that the continuous image of a connected set is connected, but I think it is not so easy to prove an interval is connected without essentially the same argument using least upper bounds; indeed that fact is essentially equivalent to the intermediate value theorem. One of my favorite definitions of connectedness is that a set S is connected iff every continuous map S-->{0,1} is constant, where {0,1} is the two point set containing only 0 and 1. It is easy to show this special case is equivalent to the full intermediate value theorem. So I could be wrong but I am afraid you are stuck with some such proof using sups. i.e. least upper bounds.
 
  • #5
Thanks for a very interesting post, mathwonk ...

Indeed, having consulted several texts on the IVT I have found, as you predict, that the proof involves sups in each case ...

Thanks again ...

Peter
 

FAQ: Intermediate Value Theorem .... Browder, Theorem 3.16 .... ....

1. What is the Intermediate Value Theorem?

The Intermediate Value Theorem is a mathematical theorem that states that if a continuous function has two values, a and b, at two different points in its domain, then it must also take on every value between a and b at some point in its domain.

2. Who discovered the Intermediate Value Theorem?

The Intermediate Value Theorem was first proved by French mathematician Bernard Bolzano in the early 19th century. However, it was later independently proven by mathematicians Augustin-Louis Cauchy and Karl Weierstrass in the mid-19th century.

3. What is the importance of the Intermediate Value Theorem?

The Intermediate Value Theorem is an important tool in calculus and real analysis. It allows us to prove the existence of solutions to equations and to show that certain functions have certain properties. It is also used in many other areas of mathematics, such as topology and differential equations.

4. Can the Intermediate Value Theorem be applied to all functions?

No, the Intermediate Value Theorem can only be applied to continuous functions. A continuous function is one that has no breaks or jumps in its graph, and can be drawn without lifting the pencil from the paper. If a function is not continuous, the Intermediate Value Theorem does not apply.

5. How is the Intermediate Value Theorem used in real life?

The Intermediate Value Theorem has many real-life applications, such as in economics, physics, and engineering. For example, it can be used to prove that a company's stock price must have been at a certain value at some point in time, or to show that a moving object must have passed through a certain position at some time. It is also used in numerical methods to find approximate solutions to equations.

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