Continuous Nowhere, Continuous Everywhere: Tricky Function Example [0,1] Domain

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Discussion Overview

The discussion revolves around finding a function that is continuous nowhere on the interval [0,1], while its square is continuous everywhere except at 0. Participants explore potential examples and engage in a broader conversation about the nature of numbers within the continuum.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant requests assistance in finding a function that meets the specified criteria of being continuous nowhere but having a square that is continuous everywhere except at 0.
  • Another participant proposes a piecewise function defined as 1 for rational numbers, -1 for irrational numbers, and 0 at x=0, suggesting it meets the criteria of discontinuity.
  • Questions arise regarding the concept of the "next number" after any given number p in the interval [0, 1], with some participants noting the difficulty in defining such a concept within a continuum.
  • Several participants assert that there is no "next number" in the set of real numbers or rational numbers, emphasizing the properties of the continuum.
  • One participant clarifies that the proposed function is indeed discontinuous at every point, as the limit does not equal the function value at any point c.
  • Another participant challenges the relevance of the "next number" question to the original function inquiry, suggesting that the function's definition does not imply any alternating behavior between rational and irrational numbers.
  • There is a consensus that the function is discontinuous at every point due to the density of rationals and irrationals in the real numbers.

Areas of Agreement / Disagreement

Participants generally agree on the discontinuity of the proposed function. However, there is disagreement regarding the implications of the "next number" question and its relevance to the original inquiry.

Contextual Notes

The discussion includes assumptions about the definitions of continuity and the properties of real numbers, which may not be universally accepted or fully explored.

Nebula
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I was asked to find a function that is continuous nowhere but its square is continuous everywhere but 0. The domain of the function is [0,1].

I can't come up with an example...
Any help is appreaciated. Thanks
 
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if your allowed to define it piecewise:

f(x)=
1 for x rational
-1 for x irrational

and if it absolutely needs to be discontinuous at 0

0 for x=0
 
StatusX said:
if your allowed to define it piecewise:

f(x)=
1 for x rational
-1 for x irrational

and if it absolutely needs to be discontinuous at 0

0 for x=0

Hmm. That raises the question: If p is a number in [0, 1], what is the "next number" after p? Is it rational or irrational? It seems rather difficult to answer for a continuum! :-)
 
One of the properties of real numbers is that there is no next number.
 
Tide said:
Hmm. That raises the question: If p is a number in [0, 1], what is the "next number" after p? Is it rational or irrational? It seems rather difficult to answer for a continuum! :-)

How on Earth did you reach that conclusion?
 
Tide said:
Hmm. That raises the question: If p is a number in [0, 1], what is the "next number" after p? Is it rational or irrational? It seems rather difficult to answer for a continuum! :-)

On the contrary, it's very easy to answer: there is NO "next number" in the set of real numbers or the set of rational numbers. If there were, every subset would be "well ordered" and that's a property of integers, not rational or real numbers.
 
matt grime said:
How on Earth did you reach that conclusion?

Actually, it wasn't a conclusion. It was a question for StatusX to ponder in which he proposes a "solution" to Nebula's query by, in effect, making "every other number" change the sign of a function. The answer, of course, is that there is no "next number" in the continuum of reals. No matter how close two numbers are you can always find other numbers in between.
 
he wanted a function that was nowhere continous, so that's what i gave. i never said that rationals and irrationals alternate, if that's what you mean, which doesn't even make sense. all i mean is that the limit of f(x) as x goes to c does not exist for any c, and so does not equal f(c), so the function isn't continuous anywhere. maybe that's not the right way to define continuity, but i still think its obviously discontinous.
 
You did have a conclusion, tide: that the example 'raised the question'... how or why does it make you think of this question? The example does not make any such 'alternating sign every other number' argument. It is just +1 and -1 for two different dense subsets. Any such function is a solution, and there's no need to put quotation marks around the word as if it weren't.

There's no next number in the rationals either, and that isn't a continuum, but the question in my mind is still, 'what has this got to do with anything?'
 
  • #10
By the definition of continuity and the fact that the rationals and irrationals are both dense in R the function is obviously discontinious as at every point.
 

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