Suppose I have a symbol/function defined pointwise like in

f[1, 5] = 3
f[3] = 7
f[2, 3.4] = 8

By executing ?f I can see the definition of f, i.e.


How can I get a list of all parameter tuples for which the function is defined? In my example I would like to obtain the list


Is that possible?


1 Answer 1


You can use DownValues[f] to get the list of rules, the structure of which uses HoldPattern to prevent the evaluation.

{HoldPattern[f[3]] :> 7, HoldPattern[f[1, 5]] :> 3, HoldPattern[f[2, 3.4]] :> 8}

If you replace f with List and extract the correct part of the expressions that should return what you are after:

(DownValues[f] /. f -> List)[[All, 1, 1, All]]
{{3}, {1, 5}, {2, 3.4}}

Note that the list has been sorted (in how DownValues reports the structure).

  • $\begingroup$ I'd use Cases[DownValues[f], HoldPattern[f[val__]] :> {val}, ∞] myself. $\endgroup$ Commented Jun 3, 2015 at 9:52
  • $\begingroup$ Great! Exactly what I need. I wasn't aware of DownValues[]. $\endgroup$
    – cgogolin
    Commented Jun 3, 2015 at 9:53
  • $\begingroup$ @Guesswhoitis, I was sure that someone would produce a smarter looking version! And that looks a lot neater than having to work out which levels to take the Part of, which is what I generally end up doing. $\endgroup$
    – SPPearce
    Commented Jun 3, 2015 at 10:01
  • 1
    $\begingroup$ @cgogolin, there are UpValues, DownValues, SubValues and OwnValues, all of which let you get at various types of definitions. $\endgroup$
    – SPPearce
    Commented Jun 3, 2015 at 10:01
  • $\begingroup$ Why do you put the last All in [[All, 1, 1, All]]? For any list f it is true that f[[All]] === f, after all. $\endgroup$
    – Jo Mo
    Commented Aug 19, 2021 at 8:53

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