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

[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!

 

[Muzza]

Construct a counterexample.

 

[HallsofIvy]

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?

 

[maverick6664]

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

Then you can prove f(x) isn't continuous : If f(x) is continuous, it must pass such a \sqrt 2 \frac m {2n} 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.