Let $f: R \in R$ be a continuous function given by $f(x+y)=f(x)+f(y)$ for all $x, y \in R$. If $\int\limits_0^2 f(x) d x=\alpha$, then $\int\limits_{-2}^2 f(x) d x$ is equal to

0

We have,

$f(x+y)=f(x)+f(y)$ for all $x, y \in R$

$\Rightarrow f(x)=\lambda x$ for all $x \in R$, where $\lambda=f$

$\Rightarrow f(x)$ is an odd function.

$\Rightarrow \int\limits_{-2}^2 f(x) d x=0$