Proving that f(x) is Not Continuous for All Real Numbers c

In summary, the function f(x) is defined as 1 if x is rational and 0 if x is irrational. This function is not continuous for all real numbers, c. However, it is continuous at x=0 and not continuous for all other real numbers c. In order to prove its discontinuity at other values, an epsilon and delta method can be used to show that for any given epsilon, there exists a neighbourhood where the difference between f(x) and f(not zero) is greater than epsilon, thus proving its discontinuity. This is due to the density property of real numbers where between every real number exists rational and irrational numbers.
  • #1
phyguy321
45
0
the function f(x) = 1 if x is rational
f(x) = 0 if x is irrational is not continuous for all real numbers, c



the function f(x) = x if x is rational
f(x) = 0 if x is irrationa is continuous at x=0 and not continuous for all other real numbers c

the function f(x) = 1/q if x is rational and x = p/q in lowest terms
f(x) = 0 if x is irrational
is continuous at c if c is irrational and not continuous at c if c is rational

I'm terrible at proofs, please help!
 
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  • #2
Hi phyguy321;! :smile:

(have an epsilon: ε and a delta: δ :smile:)

Let's start with:
phyguy321 said:
the function f(x) = x if x is rational
f(x) = 0 if x is irrationa is continuous at x=0 and not continuous for all other real numbers c

Can you use an ε and δ method to prove that it is continuous at x = 0?

Have a go. :smile:
 
  • #3
tiny-tim said:
Hi phyguy321;! :smile:

(have an epsilon: ε and a delta: δ :smile:)

Let's start with:


Can you use an ε and δ method to prove that it is continuous at x = 0?

Have a go. :smile:

I've got the proof for when f(x) is continuous at x=0 but I'm not sure how to prove that its discontinuous at every other value using delta epsilon. I understand it has to do with the density property of real numbers and that its full of holes be cause between every real number there exists rational and irrational numbers.
 

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  • #4
phyguy321 said:
I've got the proof for when f(x) is continuous at x=0 but I'm not sure how to prove that its discontinuous at every other value using delta epsilon. I understand it has to do with the density property of real numbers and that its full of holes be cause between every real number there exists rational and irrational numbers.

Hi phyguy321! :smile:

Same method … start "if x ≠ 0, then for any ε < x, … " :wink:
 
  • #5
but how do i say if |x-c|<delta then |f(x) - f(not zero)|< epsilon?
 
  • #6
phyguy321 said:
but how do i say if |x-c|<delta then |f(x) - f(not zero)|< epsilon?

But it's not continuous, so there isn't a δ.

You should be trying to prove that, no matter how small δ is, the neighbourhood will contain values that further away than ε. :smile:
 

FAQ: Proving that f(x) is Not Continuous for All Real Numbers c

What does it mean for a function to be continuous?

A function is considered continuous if it is defined at every point and the limit at every point exists and is equal to the value of the function at that point. In other words, there are no breaks or jumps in the graph of the function.

How do you prove that a function is not continuous?

To prove that a function is not continuous for all real numbers, you can use the definition of continuity and show that there exists at least one real number c for which the function is not continuous. This can be done by finding a point at which the limit of the function does not equal the value of the function at that point.

Can a function be continuous at some points and not others?

Yes, it is possible for a function to be continuous at some points and not others. A function can be continuous at all points except for a specific point or set of points where there may be a break or discontinuity.

What types of discontinuities can a function have?

There are three types of discontinuities that a function can have: removable, jump, and infinite. A removable discontinuity occurs when there is a hole in the graph of the function that can be filled to make the function continuous. A jump discontinuity occurs when there is a sudden change or jump in the graph of the function. An infinite discontinuity occurs when the limit of the function at a certain point is either positive or negative infinity.

Why is it important to prove that a function is not continuous for all real numbers c?

Proving that a function is not continuous for all real numbers c can help us understand the behavior of the function and identify any potential issues or limitations. It can also help us determine the conditions under which the function is continuous, which can be useful in solving problems and making predictions based on the function.

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