# Pointwise Addition on Real-Valued Functions is Associative

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## Theorem

Let $S$ be a set.

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

Let $f + g: S \to \R$ denote the pointwise sum of $f$ and $g$.

Then:

- $\paren {f + g} + h = f + \paren {g + h}$

That is, pointwise addition on real-valued functions is associative.

## Proof

\(\, \displaystyle \forall x \in S: \, \) | \(\displaystyle \map {\paren {\paren {f + g} + h} } x\) | \(=\) | \(\displaystyle \paren {\map f x + \map g x} + \map h x\) | Definition of Pointwise Addition of Real-Valued Functions | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map f x + \paren {\map g x + \map h x}\) | Real Addition is Associative | ||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle \map {\paren {f + \paren {g + h} } } x\) | Definition of Pointwise Addition of Real-Valued Functions |

$\blacksquare$