Function $\phi$ such that $f(\phi(x,y)) = f(x) + f(y)$

Question

I have a continuous function $f:\mathbb{R}^n\to\mathbb{R}$, and I am looking for a continuous (or at least measurable) function $\phi:\mathbb{R}^{2n}\to\mathbb{R}^n$ such that $f(\phi(x,y))=f(x)+f(y)$. This may sound like too much to demand, but at least for certain cases, it can be done easily:

• if $f$ is linear, $\phi(x,y)=x+y$,
• if $n=1$ and $f$ is bijective, $\phi(x,y)=f^{-1}(f(x)+f(y))$.

I'm interested in a much more general case — what are some necessary and/or sufficient conditions for the existence of such a function $\phi$?

Essentially, this looks like $\phi$ is a semigroup operation of some sort which is fixed by demanding that $f$ becomes a semigroup homomorphism into $\mathbb{R}$, since I'm not demanding that $\phi$ satifies the group axioms. It will have to be associative for the required identity to hold, but unless $0$ is in the image of $f$, I don't think there has to be a neutral element, nor do I think inverses need to exist unless $f$ takes both positive and negative values. On the other hand, I'm interested in continuous or at least measurable $\phi$. I'm sure there must be a theory of which kind of semigroups can be mapped into $\mathbb{R}$ in this way, but I have no idea where to start looking.

Answer 1

This is by no means complete, but too long for a comment, so I am posting some observations as a partial answer. I give some necessary conditions on $f$ for the existence of a continuous map $\phi$ under some assumptions on $f$ (to be seen later).

I. Conditions on the range $f(\mathbb{R})$

Firstly, $f(\mathbb{R}^n)$ must be closed under addition. By connectedness, this implies that the image must either be

• the singleton $\{0\}$,
• a ray of type $(-\infty, -a_0]$, $(-\infty, -a_0)$, $(+a_0, +\infty)$ or $[+a_0, +\infty)$ for $a_0 \geq 0$, or
• the entire real line $\mathbb{R}$.

In the case $f(\mathbb{R}) = \{0\}$, existence is immediate and we may take $\phi$ to be constant.

II. Group action on $\mathrm{C}(\mathbb{R}^n, \mathbb{R})$ and $\mathrm{C}(\mathbb{R}^n \times \mathbb{R}^n, \mathbb{R}^n)$

The group of homeomorphisms $\operatorname{Homeo}(\mathbb{R}^n)$ acts on $\mathrm{C}(\mathbb{R}^n, \mathbb{R})$ by precomposition and acts on $\mathrm{C}(\mathbb{R}^n \times \mathbb{R}^n, \mathbb{R}^n)$ by conjugation. Specifically, given a homeomorphism $\rho \colon \mathbb{R}^n \to \mathbb{R}^n$ and a continuous map $\phi \colon \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^n$, define $\phi^{\rho} = \rho^{-1} \circ \phi \circ (\rho \times \rho)$. Then for all $x,y \in \mathbb{R}^n$, $$ f(\phi(x, y)) = f(x) + f(y) \; \text{ if and only if }\; (f\rho)(\phi^{\rho}(x,y)) = (f\rho)(x) + (f\rho)(y). $$

Hence, there exists a continuous solution to the functional equation for $f$ if and only if there exists a continuous solution to the functional equation for any $g$ in the same orbit as $f$. For example, if $n = 1$ and $f$ is sufficiently nice, we may assume that $f$ is even piecewise linear. Similarly, by fixing either $x$ or $y$, we can make a similar conclusion on each slice of $\mathbb{R}^n \times \mathbb{R}^n$.

III. Existence of a Reeb graph homomorphism

Define $F \colon \mathbb{R}^{n} \times \mathbb{R}^n \to \mathbb{R}$ by $F(x,y) = f(x) + f(y)$. Then if $f \circ \phi = F$, it must be that $\phi$ maps level sets of $F$ into level sets of $f$. By continuity, for all $a \in \mathbb{R}$, any given connected component of $F^{-1}\{a\}$ must map into some connected component of $f^{-1}\{a\}$.

Let $\mathfrak{R}_{F} = (\mathbb{R}^{n} \times \mathbb{R}^n)/{\sim_F}$ and $\mathfrak{R}_f = \mathbb{R}^n/{\sim_f}$ be the Reeb spaces of $F$ and $f$. If we assume that $f$ and $F$ are smooth functions with finite or isolated critical values, then these spaces should¹ both assume the structure of a potentially infinite graph (with edges going out to infinity), each of which can be subdivided by including a vertex for each critical value of $F$ and $f$. Call the subdivisions $\mathfrak{R}_{F}^+$ and $\mathfrak{R}_{f}^+$ and label each vertex by the corresponding value of $F$ or $f$ at that point. Then a necessary condition for the existence of a continuous map $\phi$ should be the existence of a graph homomorphism $\Phi \colon \mathfrak{R}^+_{F} \to \mathfrak{R}^+_f$ which respects the labeling of each vertex.

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¹ It is known that for a smooth function $f \colon M \to \mathbb{R}$ on a compact manifold with finitely many critical values that the Reeb space has the structure of a graph, but I imagine there is a generalization of the mentioned sort for noncompact $M$ under potentially more restrictive conditions on $f$.