Continuity and minimum/maximums

[Brandon Taylor]

Suppose I have a function f(x), which takes some x, a real number between 0 and 2 inclusive, and returns a boolean value, true or false. Say that I also know that

f(0) = false, and
f(2) = true

I'm interested in the following two statements:

Statement A: There a minimum value of x in (0, 2] for which f will return true
Statement B: There a maximum value of x in [0, 2) for which f will return false

I suspect that one xor the other must be true, just by intuition. For example,

If f(x) = x > 1, then statement B is true but statement A is false.
If f(x) = x ≥ 1, then statement A is true but statement B is false.

I have two questions.

- Is there such way to generalize statement A to always be true? Maybe by using some kind of limit minimum? Like,

min x such that ( ∀ x' ∈ [0, 2) ( x' < x → ¬ f(x) ) )

- If not, what constraints can I put on the function f such that statement A will always be true? Maybe something about continuity?

I'm new here, so please let me know if this is the wrong kind of question for this forum.

Thanks!
Brandon

 

[Steven G]

Statement A: There a minimum value of x in (0, 2] for which f will return true
Statement B: There a maximum value of x in [0, 2) for which f will return false

I suspect that one xor the other must be true, just by intuition.

The above is not true.
What if when x= .1, .01, .001, .0001, ...., f(x) is false
If when x=1.9, 1.99. 1.999, 1.9999, ..., f(x) is true.

Which of the two statements above is true?


Is there such way to generalize statement A to always be true? Maybe by using some kind of limit minimum?
Do you mean is there a way to change A or to change f(x)?

 

[blamocur]

If for [imath]0 < x < 2[/imath] you define [imath]f(x)[/imath] as "[imath]x[/imath] is rational" then neither A nor B will be true.

 

[Brandon Taylor]

Oh, thanks, I hadn't thought about that. Hm, ok, I guess I'm going to have to put some strong restrictions on f. Maybe something like a no backsliding constraint?

¬ ( ∃ x, x' ∈ [0, 2] ( x' > x ∧ f(x) ∧ ¬ f(x') ) )

Then max x ∈ [0, 2] such that ( ∀ x' ∈ [0, 2] ( x' < x ⇒ ¬ f(x) ) )

must always exist, yes?

Is there a more concise/elegant way to say the same thing?

 

[blamocur]

¬ ( ∃ x, x' ∈ [0, 2] ( x' > x ∧ f(x) ∧ ¬ f(x') ) )

Then max x ∈ [0, 2] such that ( ∀ x' ∈ [0, 2] ( x' < x ⇒ ¬ f(x) ) )

must always exist, yes?

I believe this is correct. BTW, note that [imath]f (\max x)[/imath] can itself be either true or false.