limits - $f(x+y) = f(x)+f(y)$ continuous at $x_0=0 implies f(x)$ is continuous over R?

Let $x,y \in R$

$f(x+y) = f(x)+f(y)$

is it true that if $f$ is continuous at $x_0=0$, than $f$ is continuous in $R$?

Answer

At any arbitrary $x_1\in\mathbb{R}$ and any $\Delta\neq 0$, we have

$$f(x_1+\Delta)-f(x_1)=f(\Delta)=f(\Delta+x_0)-f(x_0).$$ As $\Delta\to 0$, the rightmost expression above goes to $0$ due to continuity at $x_0$, so the leftmost expression also goes to $0$. This implies continuity at $x_1$ and therefore in $\mathbb{R}$. Note we don't need the fact that $x_0=0$.