$\displaystyle let \ f:R \to R \ be \ continuous \ at \ some \ point \ x_0 \in R . prove $
$\displaystyle that \ if \ f(x+y) = f(x)+ f(y) , \forall x,y \in R , \ then \ f \ is \ continuous \ at \ every \ point \ of
\ R $
$\displaystyle |f(x_0 + h) - f(x_0)| = |f(x_0) + f(h) - f(x_0)| = |f(h)|$,
so given an $\displaystyle \epsilon > 0$, there is a $\displaystyle \delta > 0$ such that whenever $\displaystyle |h| < \delta$, $\displaystyle |f(h)| < \epsilon$.
Then if $\displaystyle x \in R$,
$\displaystyle |f(x + h) - f(x)| = |f(x) + f(h) - f(x)| = |f(h)|$,
so...