Is the function f(x)=1/x bounded on the interval (0,1)?

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In summary, the homework statement is that there exists a real number M such that |f(x)| ≤ M for all x in (0,1). However, the attempt at a solution does not prove that f is unbounded.
  • #1
k3k3
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Homework Statement


Let f be the function defined f(x)=1/x. Prove that f is not bounded on (0,1)


Homework Equations





The Attempt at a Solution



I think I should prove by contradiction.

Assume f is bounded on (0,1).
Since f is bounded, there exists a real number M such that |f(x)| ≤ M for all x in (0,1)
f(x) will never be negative since it is on the interval (0,1), hence |f(x)| = f(x)

This is where I begin to get unclear on where to go next. I want to show that M+1 ≤ M
Is it correct to use 1/(M+1) and plug it into f(x)?
 
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  • #2
I don't think you should prove it by contradiction. If n is an number greater than one then 1/n is in (0,1).
 
Last edited:
  • #3
Can I argue that since 1/n is an infinite sequence, then this function is not bounded?
 
  • #4
k3k3 said:
Can I argue that since 1/n is an infinite sequence, then this function is not bounded?

You need a better argument than that. What is f(1/n)?
 
  • #5
f(1/n)=n

Then I could say for all n in the positive integers?
 
  • #6
k3k3 said:
f(1/n)=n

Then I could say for all n in the positive integers?

You could say that, but it doesn't prove f is unbounded until you say why that proves f is unbounded.
 
  • #7
Since f(1/n)=n for all n in N. Since N has an infinite amount of elements, then the function is unbounded on (0,1)?
 
  • #8
k3k3 said:
Since f(1/n)=n for all n in N. Since N has an infinite amount of elements, then the function is unbounded on (0,1)?

Having an infinite number of elements has little to do with being unbounded. What does unbounded mean?
 
  • #9
That there is no lower bound, no upper bound or both.
 
  • #10
k3k3 said:
That there is no lower bound, no upper bound or both.

Ok, so give me an argument that f has no upper bound.
 
  • #11
There is no n such that 1/n is not in the interval (0,1), so there is no real number M that will satisfy |1/n|≤M.
 
  • #12
k3k3 said:
There is no n such that 1/n is not in the interval (0,1), so there is no real number M that will satisfy |1/n|≤M.

You don't want to satisfy |1/n|<=M. You want to show that you can find a number in x in (0,1) such that f(x)>M.
 
  • #13
So if M is greater than one, 1/(M+1) is in (0,1) and M+1 < M is not true?
 
  • #14
k3k3 said:
So if M is greater than one, 1/(M+1) is in (0,1) and M+1 < M is not true?

I really hope you meant f(1/(M+1))=M+1 > M.
 
  • #15
No, I was still thinking about the contradiction argument. Sorry.
 
  • #16
k3k3 said:
No, I was still thinking about the contradiction argument. Sorry.

That's ok. But you've got it now, yes?
 
  • #17
Yep. Thank you again for your help!
 

What does it mean for a function to be unbounded?

When a function is unbounded, it means that there is no limit to its values as its input variable approaches a certain value or goes to infinity.

How can you prove that a function is unbounded?

To prove that a function is unbounded, you can show that as the input variable approaches a certain value or goes to infinity, the output values also increase without limit.

Can a function be both bounded and unbounded?

No, a function cannot be both bounded and unbounded. A bounded function has a limit to its values, while an unbounded function does not.

What are some common examples of unbounded functions?

Some common examples of unbounded functions include polynomial functions with a degree greater than or equal to 2, exponential functions, and logarithmic functions.

How does knowing that a function is unbounded affect its application in real life?

Knowing that a function is unbounded can be useful in determining the behavior of the function and predicting its values in certain scenarios. It can also help in identifying potential issues or limitations in using the function for practical purposes.

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