soft question - approximately linear functions

Let $E$ and $E'$ be Banach spaces. Mappings $f:Eto E'$, which satisfy the inequality
$$|f(x + y) − f(x) − f(y)| leqepsilon$$
for all $x, y in E$, are called $epsilon$-additive (or approximately additive). The main result concerning approximately additive functions is due to D. Hyers
(link).

Theorem. Let $f(x)$ be an $epsilon$-additive mapping of a Banach space $E$ into another Banach space $E'$. Then $l(x)=lim f(2^nx)/2^n$ exists for each $x$ in $E$, $l(x)$ is linear, and the inequality $$|f(x)-l(x)|leqepsilon$$ is true for all $x$ in $E$. Moreover, $l(x)$ is the only linear mapping satisfying this inequality. If $f(x)$ is continuous at at least one point, then $l(x)$ is continuous everywhere in $E$.

So in your case $E=L^2(Omega)$, $E'=mathbb R$.

Edit. As Yemon Choi indicated, finite dimensional versions of the result had been discovered earlier and independently. Check out, for instance, the Pólya and Szegö problem book (Ch 3, Problem 99):

Assume that the terms of the sequence $a_1,a_2,a_3,dots$ satisfy the condition
$$a_m+a_n-1 < a_{m+n} < a_m+a_n+1.$$
Then
$$limlimits_{ntoinfty}frac{a_n}{n}=omega$$
exists; $omega$ is finite and we have
$$omega n-1 < a_n < omega n +1.$$