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A vacuously existing function?

  1. Mar 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Conjecture. Suppose [itex]a\in \mathbb{R}[/itex]. Suppose [itex]f[/itex] is a real-valued function defined on [itex][a,a]=\{a\}[/itex]. Suppose [itex]x\in [a,a][/itex]. Then there exists a function [itex]\phi[/itex] defined by [itex]{\displaystyle \phi(t)=\frac{f(t)-f(x)}{t-x}\quad(a<t<a,t\neq x)}[/itex].

    (i) Before proving (or disproving this) does this conjecture make sense in the first place?

    (ii) If make sense, does it truely exist?


    2. Relevant equations

    Relevant posts are:

    https://www.physicsforums.com/showthread.php?t=585386
    https://www.physicsforums.com/showthread.php?t=338366


    3. The attempt at a solution

    (i) If I kinda restate this conjecture, it becomes:

    Conjecture. Suppose [itex]a\in \mathbb{R}[/itex]. Suppose [itex]f[/itex] is a real-valued function defined on [itex][a,a]=\{a\}[/itex]. Suppose [itex]x\in [a,a][/itex]. Then there exists a function [itex]{\displaystyle \phi:\{t\in \mathbb{R}: a<t<a\} \to \mathbb{R} : t \mapsto \frac{f(t)-f(x)}{t-x}}[/itex].

    So it seems make sense in the ground of first order language and ZFC. Isn't it?

    (ii) I think this function is simply [itex]\emptyset[/itex] because the domain is empty set.
     
    Last edited: Mar 10, 2012
  2. jcsd
  3. Mar 10, 2012 #2

    jbunniii

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    There is no number [itex]t[/itex] satisfying [itex]a < t < a[/itex]. Therefore, there is no [itex]t[/itex] for which you have defined [itex]\phi(t)[/itex]. So you can say that you have vacuously created a function whose domain is the empty set. I'm not sure why it merits being called a "conjecture" or what you hope to achieve with this function.
     
    Last edited: Mar 10, 2012
  4. Mar 10, 2012 #3

    jbunniii

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    P.S. You can equally well have said the following. Let S be any set (even the empty set) and define

    [tex]\phi : \emptyset \rightarrow S[/tex]

    As the domain is empty, you don't need to specify any "formula" for how to "evaluate" [itex]\phi[/itex].

    Yes, this function exists. It is the set of points (a, s) such that [itex]a \in \emptyset[/itex] and [itex]\phi(a) = s[/itex]. Since there is no [itex]a[/itex] satisfying [itex]a \in \emptyset[/itex], the function is simply the empty set, as you indicated.
     
  5. Mar 10, 2012 #4
    Maybe I do not know clearly the meaning of conjecture. Anyway what I hope to acheive with this function is to solve some problems that I posted on this post:

    https://www.physicsforums.com/showthread.php?p=3808228&posted=1#post3808228
     
  6. Mar 10, 2012 #5

    jbunniii

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    OK, to answer your question in post #3 of that thread, yes, your quotient definition defines an empty function, and that empty function exists (in the same sense in which the empty set exists). There's no logical issue that I can see with your definition.

    It's also true that your statement in post #5 of that thread is vacuously true for any value L.

    However, the (usual) definition of a limit isn't merely that statement. I don't have Rudin here with me, so I can't check his definition, but most authors define the limit of a function at a point x as follows:

    "Let [itex]f : A \rightarrow B[/itex] be a function, and let [itex]x[/itex] be an accumulation point of [itex]A[/itex]. Then we write [itex]\lim_{t \rightarrow x} f(t) = L[/itex] if for every [itex]\epsilon > 0[/itex]..."

    i.e. the notion of a limit is defined only at accumulation points of the domain. Since the empty set has no accumulation points, the notion of a limit of an empty function is undefined.
     
  7. Mar 13, 2012 #6
    Yes, what you say is exactly true and I agree with this to a full extent. But if you read my post carefully, you will see that the problem I proposed arises exactly because of what you said, that is, because the notion of limit is undefined at a point which is not a limit point, I can't use Rudin's definition to get the deriviative of a function defined at a singleton. (The problem is not of limit-definition but derivative-definition.)

    Anyway as for this post, please answer to that post.

    You know, all I wanna get is a proper definition of derivative that I can use on a function defined on a singleton (at an isolated point). But maybe (what I'm doing) it's something time-wasting probably because general mathematicians wouldn't care about the derivative of a function at an isolated point.
     
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