Does continuity at a point and Additive function imply continuity at all other points in a normed linear space. Is there some result like there exist a in field such that f(x) = ax for all x in normed linear space
Motivation: f additive dunction from R to R and continuous at some a. Then f(x)=f(1)*x for all x in R. And is this result in R dependent on Axiom of Countable choice or we can prove directly also
The statement about continuity is true: If $f$ is additive, the difference $f(x+h)-f(x)$ is independent of $x$, hence continuity can be tested at one single point.
Furthermore if $f$ is defined on the real numbers, additive and continuous at a single point, it is continuous everywhere, hence determined by the values $f(x), x \in \mathbb Q$. In particular it is linear.