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][1])

> **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 $\R^n$.


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=On%20the%20stability%20of%20the%20linear%20functional%20equation&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq