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Consider a partial function $f$ that is defined only for a few values of its domain (my exact use case is $\delta$ transition functions in automata). One can 'complete' it by saying $$g(x)=0\iff f(x) \text{ is not defined.}$$

Is there a symbol to mean "undefined"? Would it be correct, or accurate, to write $\nexists f(x)$?

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    $\begingroup$ Some people write $f(x)\uparrow$. $\endgroup$
    – Chris Eagle
    Commented Sep 4, 2011 at 21:57
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    $\begingroup$ I remember $\bot$ being used. See en.wikipedia.org/wiki/Partial_function#Bottom_type. $\endgroup$
    – Srivatsan
    Commented Sep 4, 2011 at 21:59
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    $\begingroup$ However $\not\exists f(x)$ seems confusing though. $\endgroup$
    – Srivatsan
    Commented Sep 4, 2011 at 22:02
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    $\begingroup$ @Srivatsan You mean, $f(x) = \bot$? I remember something of the sort. $\endgroup$
    – badp
    Commented Sep 4, 2011 at 22:09
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    $\begingroup$ Ah yes. I meant $f(x) = \bot$. (But this is just what I remember, so not to be trusted. Hopefully some expert can corroborate.) $\endgroup$
    – Srivatsan
    Commented Sep 4, 2011 at 22:10

2 Answers 2

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A language for mathematical knowledge management uses $f(x)\uparrow$.

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    $\begingroup$ I had not seen this before, but I have seen $f(x) \downarrow$ to mean that $f(x)$ is defined, so this makes sense. $\endgroup$
    – Toby Bartels
    Commented Jan 22, 2016 at 19:19
  • $\begingroup$ The link you provided is dead :( $\endgroup$
    – exhuma
    Commented Mar 6, 2019 at 9:44
  • $\begingroup$ @exhuma I replaced it with a more stable link. $\endgroup$
    – Charles
    Commented Mar 6, 2019 at 13:22
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    $\begingroup$ I think you will loose more readers using this symbol rather than not using it and writing in plain text. From what I have seen on this site purely logical symbols are not commonly used except in this field (logic) and the large majority of people replace them by plain text (and/or for instance are almost never replaced by $\land,\lor$ in general math). $\endgroup$
    – zwim
    Commented Apr 19, 2020 at 21:22
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Thus, $f(x)\uparrow$ and $f(x)=\bot$ were suggested, although the prior use of the latter is perhaps not common (I did not quickly find it from Google).

$f(x)=\frac00$ might suit some contexts.

I would use none of the above without defining it first.

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