Counterexample: Showing the Non-Continuity of a Function with Rational Values

In summary, it has been shown that if f(x) is defined for all x in [a,b] with f(b) > f(a) [values given] and the values of f at any x in (a,b) is rational, then f(x) is not continuous. This can be proven by finding an irrational number between f(a) and f(b) and showing that f(x) cannot pass through it if it is continuous.
  • #1
loli12
I was asked to show whether this is true: f(x) is defined for all x in [a,b] with f(b) > f(a) [values given]. the values of f at any x in (a,b) is rational. So, is f(x) continous?

I think this is not continuous as this seems like the question is trying to use intermediate value property to imply continuity. But I can;t think of a more proper way to proof the answer. Or am I wrong on this? Please give me some idea!
 
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  • #2
Construct a counterexample.
 
  • #3
loli12 said:
I was asked to show whether this is true: f(x) is defined for all x in [a,b] with f(b) > f(a) [values given]. the values of f at any x in (a,b) is rational. So, is f(x) continous?
I think this is not continuous as this seems like the question is trying to use intermediate value property to imply continuity. But I can;t think of a more proper way to proof the answer. Or am I wrong on this? Please give me some idea!


"this seems like the question is trying to use intermediate value property to imply continuity." I would have said it the other way around! If the function were continuous, then it would have the intermediate value property. Whatever f(a) and f(b) are, since f(b)> f(a) they are not the same. Does there exist an irrational number between them? What does that tell you?
 
  • #4
loli12 said:
I was asked to show whether this is true: f(x) is defined for all x in [a,b] with f(b) > f(a) [values given]. the values of f at any x in (a,b) is rational. So, is f(x) continous?
I think this is not continuous as this seems like the question is trying to use intermediate value property to imply continuity. But I can;t think of a more proper way to proof the answer. Or am I wrong on this? Please give me some idea!

Right. It's not continuous. You can prove there is (are) at least an irrational number between f(a) and f(b).

Let's use [tex]\sqrt 2 < 2[/tex] as the irrational number. Put A = f(a) and B = f(b). You can find a natural number n which satisfies [tex] B - A < \frac 1 n[/tex] Then, at least one of[tex]\sqrt 2 \frac m {2n} [/tex] (where m is an integer) must be between A and B. (notice step of [tex]\sqrt 2 \frac m {2n}[/tex] is smaller than B-A.)

Then you can prove f(x) isn't continuous : If f(x) is continuous, it must pass such a [tex]\sqrt 2 \frac m {2n}[/tex] which is evidently an irrational number.

BTW you can easily prove the opposite theory: If f(x) takes only irrational numbers between [a,b], then f(x) cannot be continuous.
 
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FAQ: Counterexample: Showing the Non-Continuity of a Function with Rational Values

What is a counterexample in science?

A counterexample in science is an example or situation that contradicts a hypothesis, theory, or generalization. It is used to disprove or challenge a proposed explanation or idea.

How do you construct a counterexample?

To construct a counterexample, you need to identify the hypothesis or theory that you want to challenge and then find an example or situation that goes against it. This can involve conducting experiments, collecting data, and analyzing results.

Why is it important to construct counterexamples?

Constructing counterexamples is important because it allows scientists to test and refine their theories and hypotheses. It also helps to identify any flaws or limitations in current scientific knowledge and can lead to new discoveries and advancements.

Can a counterexample ever be proven wrong?

No, a counterexample cannot be proven wrong because it is specifically designed to challenge or disprove a hypothesis or theory. However, it can be further examined and potentially used to revise or refine the existing explanation.

Are there any limitations to using counterexamples in science?

While counterexamples can be powerful tools in science, they also have limitations. For example, a counterexample may not always be applicable to every situation or may not fully disprove a theory. Additionally, constructing counterexamples can be time-consuming and may not always lead to definitive conclusions.

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