# Definition:Pointwise Addition of Real-Valued Functions

## Definition

Let $S$ be a non-empty set.

Let $f, g: S \to \R$ be real-valued functions.

Then the pointwise sum of $f$ and $g$ is defined as:

$f + g: S \to \R:$
$\forall s \in S: \map {\paren {f + g} } s := \map f s + \map g s$

where the $+$ on the right hand side is real-number addition.

Thus pointwise addition is seen to be an instance of a pointwise operation on real-valued functions.