The sum of functions is also defined pointwise . Therefore, whatever is true for individual points also holds for the whole function. Associativity is shown in a similar manner. It's worth noting that pointwise and componentwise definitions are actually the same. If we do, the pointwise function addition will be reduced to the componentwise addition of vectors.
Notations reinforce the analogy.
X Y is denoted Y X . These functions take on only values 0 and 1 . Thus there is a 1-1 correspondence between subsets of X and their characteristic functions. For this reason the set of all subsets of X is often denoted as 2 X . It must also be noted that functions become interesting when the spaces X and Y topological .