Suppose $f(x)$ is defined on the entire line and continuous at the origin with the property that $f(\alpha+\beta) = f(\alpha) + f(\beta)$ where $\alpha,\beta \in \mathbb{R}$. Prove that $f(x)$ must be continuous at every point $x=a$.
Try:
First notice that $f(0) = f(0+0)=f(0)+f(0)=2f(0) \implies f(0)=0$. Since $f$ continuous at zero, then $\lim_{x \to 0} f(x) = f(0) = 0$. Next, let $a $ be arbitary real number. Then,
$$ \lim_{ x \to a} f(x) = \lim_{x \to a} f(x-a)+f(a)=_{h = x-a} \lim_{h \to 0} f(h) + f(a) = 0 + f(a)=f(a)$$
So f is continuous everywhere.
Is this a correct solution?
