Intermediate Value Theorem Converse

In summary, the problem is determining if the converse of the Intermediate Value Theorem in the given book is true. The attempted solution involved looking at examples and considering the continuity of the function. The converse of the theorem would state that if there is at least one number c in the closed interval [a,b] such that f(c)=k, then f is continuous on [a,b] and k is any number between f(a) and f(b). The conclusion is that the converse is possibly false.
  • #1
KF33
19
0

Homework Statement


I was given the problem of determining if the Converse of the Intermediate Value Theorem in my book was true. Below is my theorem from the book.

Homework Equations

The Attempt at a Solution


I had looked at the converse and tried to draw some examples, and I am thinking it is false. I am leaning that way, because technically the function may or may not be continuous. I just need to know if I am on the right direction.
 

Attachments

  • card-26189841-back.jpg
    card-26189841-back.jpg
    21 KB · Views: 507
Physics news on Phys.org
  • #2
KF33 said:

Homework Statement


I was given the problem of determining if the Converse of the Intermediate Value Theorem in my book was true. Below is my theorem from the book.

Homework Equations

The Attempt at a Solution


I had looked at the converse and tried to draw some examples, and I am thinking it is false. I am leaning that way, because technically the function may or may not be continuous. I just need to know if I am on the right direction.

What, exactly, would be the converse of the intermediate-value theorem?
 
  • #3
Ray Vickson said:
What, exactly, would be the converse of the intermediate-value theorem?
If there is at least one number c in [a,b] such that f(c)=k, then f is continuous on the closed interval [a,b] and k is any number between f(a) and f(b). I got the answer though I think.
 

FAQ: Intermediate Value Theorem Converse

What is the Intermediate Value Theorem Converse?

The Intermediate Value Theorem Converse is a mathematical theorem that states that if a function is continuous between two points, then it must take on every value between those two points.

What is the importance of the Intermediate Value Theorem Converse?

The Intermediate Value Theorem Converse is important because it allows us to determine if a function has a certain value between two points without having to actually calculate the value. This is especially useful when dealing with complicated functions.

What is the difference between the Intermediate Value Theorem and its Converse?

The Intermediate Value Theorem states that if a function is continuous between two points, then it must have a value between those two points. The Converse states that if a function has a value between two points, then it must be continuous between those two points.

How is the Intermediate Value Theorem Converse used in real life applications?

The Intermediate Value Theorem Converse has many practical applications, such as in economics, physics, and engineering. For example, it can be used to determine the existence of solutions to real-world problems, such as finding the root of an equation or the maximum or minimum value of a function.

Are there any limitations to the Intermediate Value Theorem Converse?

Yes, there are limitations to the Intermediate Value Theorem Converse. It only applies to continuous functions, and it does not provide an exact value for the function. It only guarantees the existence of a value between two points.

Similar threads

Using the Intermediate Value Theorem to Find Fixed Points
Replies
2
Views
1K
Let function ƒ be Differentiable
Replies
2
Views
1K
Intermediate Value Theorem Problem on a String
Replies
14
Views
2K
Intermediate Value Theorem question
Replies
8
Views
2K
Egoroff's Theorem: Finite Measure Convergence
Replies
3
Views
2K
Proving Bounded Function and Continuity: Intermediate Value Theorem Explained
Replies
6
Views
2K
Question on Mean Value Theorem & Intermediate Value Theorem
Replies
3
Views
1K
Final Value Theorem Rule Clarification
Replies
2
Views
2K
Extreme and Intermediate value theorem
Replies
2
Views
2K
Finding the Average Value of a Function
Replies
13
Views
2K

Hot Threads

Recent Insights

Back
Top