Functions , which satisfy  the inequality
$$|f(x + y) − f(x) − f(y)| \leq\epsilon$$
for all $x, y \in \mathbb R^n$, satisfy are called $\epsilon$-additive (or approximately additive). The main result concerning approximately additive functions is due to D. Hyers

> **Theorem.** Let $f: \mathbb R^n \to \mathbb R$ be an  $\epsilon$-additive function. Then there is a unique additive function $g: \mathbb R^n \to \mathbb R$ such that
$$|f(x)-g(x)|\leq\epsilon $$
for all $x\in \mathbb R^n$.