# Possible Unique Function Math

**URL:** <https://boards.straightdope.com/t/possible-unique-function-math/946363>\
**Category:** Factual Questions\
**Created:** [July 13, 2021, 2:49am UTC](https://boards.straightdope.com/t/possible-unique-function-math/946363 "2021-07-13T02:49:00Z")\
**Posts on this page:** 4\
**Page:** 2

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 14, 2021, 10:56pm UTC](https://boards.straightdope.com/t/possible-unique-function-math/946363/21 "2021-07-14T22:56:39Z")

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Now that I think about it, while on one hand you can patch together/interpolate (not uniquely!) an analytic function by specifying its expansion at a discrete set of points, on the other hand on _compact_ Riemann surfaces like the Riemann sphere there are very strong geometric constraints given by the Riemann–Roch theorem, or even just consider Liouville’s theorem that if a function is holomorphic on the entire Riemann sphere then you already know it must be constant.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [July 15, 2021, 3:57pm UTC](https://boards.straightdope.com/t/possible-unique-function-math/946363/22 "2021-07-15T15:57:13Z")

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> [@DPRK](#):
>
> Now that I think about it, while on one hand you can patch together/interpolate (not uniquely!) an analytic function by specifying its expansion at a discrete set of points

I was thinking about this. What about the following situation? Let D be the set \{1, 2, 2 1/2, 3, 3 1/3, ..., n, n+1/n,...\}. Then D is certainly discrete. Let the function f be 1 at integers and 0 at non-integers. Can this be extended to an analytic function? I do not know the answer to this question. On the other hand, say that a set is _uniformly discrete_ if there is a lower bound to the distant between any two points. Then I am certain your conclusion is correct.

Actually, the set \{1, 1/2, 1/3, 1/4, ...\} is also discrete and you can’t even get a continuous function that interpolates the function that alternates between 1 and 0. So you will certainly require not only discrete but also closed. A uniformly discrete set is obviously closed.

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**Author:** ![DPRK](https://avatars.discourse-cdn.com/v4/letter/d/4491bb/32.png) [@DPRK](https://boards.straightdope.com/u/DPRK)\
**Post date:** [July 16, 2021, 12:44am UTC](https://boards.straightdope.com/t/possible-unique-function-math/946363/23 "2021-07-16T00:44:31Z")

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I was clearly being sloppy when I typed “discrete”, as your example

\{1,\,\frac{1}{2},\,\frac{1}{3},\,\ldots\}

shows.

Now eg Theorem 26.7 in Forster’s Lectures on Riemann Surfaces shows that if X is a non-compact Riemann surface and a\_1,\, a\_2,\,a\_3, \,\ldots is a sequence of distinct points on X which has no accumulation point, then given arbitrary complex numbers c\_n\in\mathbb{C} there exists a holomorphic function on X with f(a\_n) = c\_n.

In particular, your set \{1,\,2,\,2\frac{1}{2},\,3,\,3\frac{1}{3},\,\ldots\} is not problematic; it does not have to be “uniformly discrete”, merely discrete and closed.

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**Author:** ![Hari\_Seldon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/hari_seldon/32/5173_2.png) [@Hari\_Seldon](https://boards.straightdope.com/u/Hari_Seldon)\
**Post date:** [July 16, 2021, 4:21pm UTC](https://boards.straightdope.com/t/possible-unique-function-math/946363/24 "2021-07-16T16:21:16Z")

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I assume you mean that given an arbitrary sequence \{c\_n\in\mathbb C\},…

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