In this question there was raised a question on general sufficient conditions under which we cannot factor the functions and there is an answer which states that:
Theorem: Let $f:\mathbb R^2\to\mathbb R$. There exist $g,h:\mathbb R\to\mathbb R$ such that $f(x,y) = g(x)h(y)$ if and only if for all $x,x',y,y'\in\mathbb R$ we have $f(x,y)f(x',y')=f(x,y')f(x',y)$.
This gives us necessary and sufficient conditions for functions of two variables.
When I was thinking a little bit about this question and the answer the natural question came on the form which functions that can be factored take.
The only continuous functions that can be factored which I was able to find are these functions (let us work with two variables):
1) The functions of the form $f(x,y)=ax^iy^j$ where $a$ is some real number and $i,j \in \mathbb N_0$.
2) The functions of the form $f(x,y)=b^{g(x,y)}$ where $b \in \mathbb R^+ \cup \{0\}$ and $g(x,y)$ is some function which have terms that depend either on $x$ alone or on $y$ alone, but not both on $x$ and $y$, for example $g(x,y)=x^2+x+y$ would work, the $g(x,y)=x+ \frac {1}{y}$ would also work, but $g(x,y)=x+y+xy$ would not work.
My question is:
Are there some continuous functions of two variables which are not of the two above stated forms but which can be factored?
Remark: I know that the words with which I described $g(x,y)$ are more descriptive than rigorous but at the moment I do not know how to rigorously define such functions which have terms that are dependent either on the first or on the second variable but not on both, but I believe that you understand.
No, (1) and (2) do not capture all such functions. E.g., $f(x,y)=x+1$ fits neither form (functions of form (2) do not change sign).
The simplest form that all such functions take is $f(x,y)=g(x)h(y)$. Yes, it's a tautology, but the best you can get. Note that your form (2), despite not capturing all such functions, is already not much simpler than the above, since it involves decomposing $\log f$ or $\log (-f)$ into a sum of two functions.
One useful thing that comes from considering $\log f$ is the differential equation $$ \frac{\partial ^2}{\partial x \partial y} \log f = 0 \tag{1} $$ which characterizes the functions that are the sum of a function of $x$ and a function of $y$. It leads to $$ f f_{xy} - f_xf_y=0 \tag{2} $$ Unlike (1), the equation (2) does not assume positivity (or constant sign), and provides a description of all sufficiently smooth functions such that $f(x,y)=g(x)h(y)$ by means of a PDE (a quite degenerate PDE, unfortunately).