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When I'm explaining things involving partial functions, I usually end up stumbling over my words, like so: "Suppose $f : A \rightarrow B$ is a function, uhh, sorry I mean a partial function, and suppose $S$ denotes its support. Then there's a corresponding restriction function... um, well, really its a partial function, $\overline{S} : A \rightarrow A$. And $S$ is characterized as the smallest set such that $f \circ \overline{S}=f.$"

I find that initially, I intentionally correct for this, but after awhile, "relapse" occurs, maybe I'll call $\frac{1}{x}$ a function from the real line back to itself, then add something about "well, really its a partial function" etc., and pretty soon the explanation starts to get bogged down in these sorts of tiresome qualifications. I'll bet other people experience this, too.

Anyway, I was just wondering if there's any accepted single-word that means "partial function," since this would kind of solve the problem.

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    $\begingroup$ Just call it a punction and accept it. $\endgroup$
    – Mikhail Katz
    Commented Mar 8, 2017 at 15:55
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    $\begingroup$ This frequently occurs when discussing computability. I don't think there's a single word that works, but I can suggest the following workaround against forgetting the adjectives: declare that "functions" are renamed to "total functions" and that any occurrence of "function" must be qualified by "partial" or "total"—and that if you ever forget to use one or the other, the audience should interrupt you if there is the slightest ambiguity. $\endgroup$
    – Gro-Tsen
    Commented Mar 8, 2017 at 15:58
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    $\begingroup$ Also, I recommend the notation $X\dashrightarrow Y$ (with a dotted / dashed arrow) to represent partial functions, rather than $X\to Y$ or even $X\rightharpoonup Y$, it's more visually distinctive. $\endgroup$
    – Gro-Tsen
    Commented Mar 8, 2017 at 16:00
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    $\begingroup$ I disagree that $X ⇢ Y$ is visually more distinctive than $X \rightharpoonup Y$. $\endgroup$
    – Andrej Bauer
    Commented Mar 8, 2017 at 20:14
  • $\begingroup$ Some use an arrow with a vertical stroke to denote a partial function and distinguish it visually from an (ordinary, total) function. I myself find it very convenient, and use it regularly when dealing with induced densities. I don't know if there is a built-in tag in the current distributions of LaTeX for something like that, but just in case you may use (a variant of) the following (after replacing each section sign "§" with a dollar): \newcommand\pto{\mathrel{\ooalign{\hfil§\mapstochar§\hfil\hfil\cr§\to§\cr}}} $\endgroup$
    – Salvo Tringali
    Commented Mar 8, 2017 at 20:47

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