# Math Help - analysis!!

1. ## analysis!!

$let \ f:R \to R \ be \ continuous \ at \ some \ point \ x_0 \in R . prove$
$that \ if \ f(x+y) = f(x)+ f(y) , \forall x,y \in R , \ then \ f \ is \ continuous \ at \ every \ point \ of
\ R$

2. Originally Posted by flower3
$let \ f:R \to R \ be \ continuous \ at \ some \ point \ x_0 \in R . prove$
$that \ if \ f(x+y) = f(x)+ f(y) , \forall x,y \in R , \ then \ f \ is \ continuous \ at \ every \ point \ of
\ R$
$|f(x_0 + h) - f(x_0)| = |f(x_0) + f(h) - f(x_0)| = |f(h)|$,

so given an $\epsilon > 0$, there is a $\delta > 0$ such that whenever $|h| < \delta$, $|f(h)| < \epsilon$.

Then if $x \in R$,
$|f(x + h) - f(x)| = |f(x) + f(h) - f(x)| = |f(h)|$,
so...