Does a functional equation $f : R → R$ include all solutions of $f : N → N$

Question

If I have a functional equation defined in $f : N → N$ and I have to show that there are no solutions. If I show it for $f : R → R$, does it directly imply there is no solution in $f : N → N$?

Question part 2: even if I would substitute 0 for some variable and the result would show there are no solutions in $f : R → R$, does that mean there are no solutions in $f : N → N$? Seems odd to me

Can someone confirm? Seems tirvial, but the problem Im solving seems to easy that way. — Csaba Daniel Farkaš, May 20 '20 at 17:07
What precisely do you mean by "a functional equation $f: N \to N$"? $f$ is a function, not a functional equation. — Robert Israel, May 20 '20 at 17:14
In the whole text I mean defined in. But thanks for pointing it out. — Csaba Daniel Farkaš, May 20 '20 at 17:16
But you still didn't say precisely what a functional equation is. — Robert Israel, May 20 '20 at 17:18
Clearly @Danjel means a function that is expressed as an equation. — David G. Stork, May 20 '20 at 17:19

score 1 · Accepted Answer · answered May 20 '20 at 17:25

1

For example, the "functional equation" $f(x)^2 = \cos(2\pi x)$ has solutions for functions $\mathbb N \to \mathbb N$, but not for $\mathbb R \to \mathbb R$.

answered May 20 '20 at 17:25

Robert Israel

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Does a functional equation $f : R → R$ include all solutions of $f : N → N$

1 Answers1