Additively separable function

Definition

For a function of two variables

Suppose $F$ is a function of two variables $x$ and $y$. We say that $F$ is additively separable if there exist functions $f,g$ of one variable such that:

$F(x,y)=f(x)+g(y)$

on the entire domain of $F$.

Note that the concept of additively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally additively separable need not remain additively separable.

For a function of many variables

Suppose $F$ is a function of $n$ variables $x_1,x_2,\dots ,x_n$. We say that $F$ is completely additively separable if there exist functions $f_1,f_2,\dots ,f_n$, each a function of one variable, such that:

$F(x_1,x_2,\dots ,x_n)=f_1(x_1)+f_2(x_2)+\dots +f_n(x_n)$

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially additively separable: if we express the set $\{1,2,\dots ,n\}$ as a union of two disjoint subsets $A,B$, $F$ is additively separable with respect to the partition if there exist functions $f_A,f_B$ such that:

$F(x_1,x_2,\dots ,x_n)=f_A(\text{only the variables }{x}_i,i\in A)+f_B(\text{only the variables }{x}_i,i\in B)$

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Partial derivatives

Additively separable functions are the exceptions to the general rule that value of partial derivative depends on all inputs.

Version type	Statement about first-order partial derivatives	Statement about second-order mixed partial derivatives
additively separable function $F$ of two variables $x,y$, both pieces are differentiable functions, written as $F(x,y)=f(x)+g(y)$	$F_x(x,y)=f^{\prime }(x)$ (independent of $y$) $F_y(x,y)=g^{\prime }(y)$ (independent of $x$)	$F_{xy}(x,y)=0$ $F_{yx}(x,y)=0$
completely additively separable function $F$ of $n$ variables $x_1,x_2,\dots ,x_n$, written as $f_1(x_1)+\dots +f_n(x_n)$	$F_{x_i}(x_1,x_2,\dots ,x_n)={f_{i^{\prime }}}^{\prime }(x_i)$ for each $i$. Note that each first-order partial depends only on that variable and not on the others.	$F_{x_i{x}_j}(x_1,x_2,\dots ,x_n)=0$ for each $i,j$.
partially additively separable function $F(x_1,x_2,\dots ,x_n)=f_A(\text{only the variables }{x}_i,i\in A)+f_B(\text{only the variables }{x}_i,i\in B)$	Each first-order partial of $F$ with respect to a variable in $A$ equals the corresponding first-order partial of $f_A$, and in particular depends only on the variables within $A$. Each first-order partial of $F$ with respect to a variable in $B$ equals the corresponding first-order partial of $f_B$, and in particular depends only on the variables within $B$.	Any second-order mixed partial involving a variable in $A$ and a variable in $B$ is zero.