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Is it allowed to define local functions inside the Module in Mathematica? For example, out of the two codes below which one is correct?

First:

plt1[a_, b_] := 
    Module[{x, y},
        f[x_, y_] := Sin[x*y];
        Plot3D[f[x, y], {x, -a, a}, {y, -b, b}]
    ];


plt1[1, 2]

Second:

plt2[a_, b_] := 
    Module[{f},
        f[x_, y_] := Sin[x*y];
        Plot3D[f[x, y], {x, -a, a}, {y, -b, b}]
    ];


plt2[1, 2]
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    $\begingroup$ The first one makes x,y local, but f global. Making x,y, local is not necessary, because Plot3D localizes its variables. The second one makes f local. Therefore, the second one is the right way. $\endgroup$ Jan 8, 2022 at 14:15
  • $\begingroup$ @Daniel Your point is made very clearly in your comment. Perhaps you could convert it to an answer. $\endgroup$
    – MarcoB
    Jan 8, 2022 at 17:57
  • $\begingroup$ @Daniel Thanks a lot for the clarifications. $\endgroup$
    – Harken
    Jan 8, 2022 at 22:12
  • $\begingroup$ You have to be careful with local functions like your f in the second example, since often (but not in your case above) they will not be automatically garbage-collected. I recommend reading this Q/A for more information on that. $\endgroup$ Jan 10, 2022 at 16:48

2 Answers 2

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The first example makes x,y local, but f global. Making x,y, local is not necessary, because Plot3D localizes its variables. This leaves f global what can creates troubles.

The second example makes f local and does not unnecessarily localize x and y. Therefore, the second example is the right way.

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Yes. Have a look at the localized definition of a function using a localized symbol.

Module[{f}, f[x_] := x^2; DownValues[f]]
(* {HoldPattern[f$6838[x_]] :> x^2} *)

Module renames the localized symbols. In this case f becomes f$6838. You may do the same things with a localized symbol that you can do with a global symbol.

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