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