If a function such that $f(x+y)=f(x)+f(y)$ is continuous at $0$, then it is continuous on $\mathbb R$

Question

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$. If $f$ is continuous at zero how can I prove that is continuous in $\mathbb{R}$.

Answer 1

Prove it is Lipschitz. (Actually you can prove it is of the form $f(x) = cx$ for some constant $c$.) See also this thread.

Answer 2

Hint: Use the definition, prove that $f(0)=0$ and use $$f(a+\varepsilon)-f(a)=f(\varepsilon)\,.$$

Answer 3

1. Show $f(0)=0$
2. For any $x\in\mathbb{R}$, $|f(x+h)-f(x)|=|f(h)|$