Additively separable function
Contents
Definition
For a function of two variables
Suppose is a function of two variables and . We say that is additively separable if there exist functions of one variable such that:
on the entire domain of .
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 is a function of variables . We say that is completely additively separable if there exist functions , each a function of one variable, such that:
(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 as a union of two disjoint subsets , is additively separable with respect to the partition if there exist functions such that:
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 firstorder partial derivatives  Statement about secondorder mixed partial derivatives 

additively separable function of two variables , both pieces are differentiable functions, written as  (independent of ) (independent of ) 

completely additively separable function of variables , written as  for each . Note that each firstorder partial depends only on that variable and not on the others.  for each . 
partially additively separable function equals  Each firstorder partial of with respect to a variable in equals the corresponding firstorder partial of , and in particular depends only on the variables within . Each firstorder partial of with respect to a variable in equals the corresponding firstorder partial of , and in particular depends only on the variables within . 
Any secondorder mixed partial involving a variable in and a variable in is zero. 