Defining a function which is numerical on all vectors of arguments

There is a slight problem with NumericQ, that prevents one to define your UpValues with it. So that would not work as expected:

f /: NumericQ[_f] := True

But manually

UpValues[f] = {HoldPattern[NumericQ[_f]] :> True};

it works as expected:

NumericQ[f[{1, 2, 3, 4}]]
NumericQ[f[1, {2}, 3, {4}]]
(* True *)
(* True *)

I thought @swish would incorporate my comment, but let me put it here, since it is an answer to the question. In the following, the arguments of f can be numbers or any sort of nested lists of numbers.

f /: NumericQ[f[args___]] /; VectorQ[Flatten@{args}, NumericQ] := True;

NumericQ[f[1, 2, 3, 4]]
NumericQ[f[{1, 2, 3, 4}]]
NumericQ[f[1, {2}, 3, {4}]]
NumericQ[f[x]]  (* False -- should check both cases T, F *)
NumericQ[f[]]   (* True  -- same as `NumericQ[Sin[]]` *)
(*
  True
  True
  True
  False
  True
*)

---

Update

@pisco points out that the above does not work if f[..] is nested inside other numeric expressions. There is an internal function that could be used to replace NumericQ:

ClearAll[f];
Attributes@f = {NumericFunction};

Internal`WouldBeNumericQ[2 + f[1, {2}, 3], {}]
(*  True  *)

The last list is a list a variables, the function tests whether f would be numeric if numeric values were substituted for the variables.