Let $f$ be a real valued function defined on $\mathbb R$ such that $f(x+y)=f(x)+f(y)$. Suppose there exists at least an element $x_0 \in \mathbb R$ such that $f$ is continuous at $x_0$. Then prove that $f(x)=ax$, for some $x \in \mathbb R$.
Hints will be appreciated.
By the Cauchy equation $f(x) = ax$ for some $a \in \mathbb R$ and all $x \in \mathbb Q$.
Then proof that if $f$ is continuous at a point, it is continuous everywhere.
Then use the continuity to conclude that $f(x) = ax$ holds for all $x \in \mathbb R$. Note that there is always a rational number between two reals.
