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

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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 $\mathbb{R}$.

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• By sliding around a neighborhood of zero. In other words if $\delta$ is very close to zero, then by continuity, so is $f(\delta)$, and so is $f(x + \delta) = f(x) + f(\delta)$. Now just formalize that argument. – user4894 Oct 20 '14 at 19:59
1. Show $f(0)=0$
2. For any $x\in\mathbb{R}$, $|f(x+h)-f(x)|=|f(h)|$
Prove it is Lipschitz. (Actually you can prove it is of the form $f(x) = cx$ for some constant $c$.) See also this thread.
Hint: Use the definition, prove that $f(0)=0$ and use $$f(a+\varepsilon)-f(a)=f(\varepsilon)\,.$$