andydude Wrote:So, not only can we show that f(x) is a constant, but we can show it in multiple ways, which is a very bad sign. I think this is proof that no such function can exist (or the only solution to all of these equations is f(x)=1).

Yeah, I see its even much simpler to prove than I did, and we can also drop the demand of continuity. However your proof(s) omits the case of . So I will add the detail:

If for f being defined on , then for . So if there is at least one with (i.e. is not identically 0) then .

Let now in our original equation: . With we get for all .

So either or .

@Ivars: this proof also works for hyperreals.