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Does there exist an elementary function on some subset $I$ of $R$ such that you can prove that $A = \{\sup(f(x)): x \in I\}$ exists, but you cannot find the value of $A$ ?

If the answer is "no", then replace "elementary function" above by "function" as a further question

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    Does something like $A = \sup \{1 - \frac 1x \mid x \in 2\mathbb Z, x \ge 4, \exists p,q \text{ prime }: x = p+q\}$ count for you?2012-11-06
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    Ha. Good points. Will have a think about that and get back to you later.2012-11-06
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    Martini - your comment only makes sense if you are able to prove that A exists. Which you are not. Will - what is meant by "cannot find" is self-evident. And I don't see any problems with doing this.2012-11-07
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    Bumpety bump bump2012-11-12
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    Ah..... A exists because we do know that {x: x is even, x = p+q where p and q are prime} is unbounded.2012-11-13
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    Now I also see that "cannot find" is relevant. If Goldbach's conjecture were solved with the expected result being proven true then martinin's example wouldn't count. But the fact that the Goldbach's conjecture hasn't been solve does not mean that we will not be able to find A in the future. I want an example whereby you can a) prove that A exists, b) prove that it is impossible to find A. So for example you could show that -1 is a lower bound and 0 is definitely not a lower bound, and you can prove that finding the exact value of A is an impossible task.2012-11-13
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    I guess the answer is no, because you can probably always use some computer programme to split the range into intervals of 2, then 4, then 8, and and so on, narrowing down the supremum or infimum to desired accuracy.2012-11-13

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