Show a function for which $f(x + y) = f(x) + f(y) $ is continuous at zero if and only if it is continuous on $\mathbb R$ 
Suppose that $f: \mathbb R \to\mathbb  R$ satisfies $f(x + y) = f(x) + f(y)$ for each real $x,y$.
Prove $f$ is continuous at $0$ if and only if $f$ is continuous on $\mathbb R$.

Proof: suppose $f$ is continuous at zero. Then let $R$ be an open interval containing zero. Then $f$ is continuous at zero if and only if $f(x) \to f(0)$ as $x \to 0$. Then $|f(x) - f(0)| < \epsilon$.
Can anyone please help me? I don't really know how to continue. Thank you.

Hints:
If $f(x)$ is continuous at 0, then we have $\lim_{x\to 0}f(x)=0$. 
Then $\lim_{x\to 0}f(x_0+x)=\lim_{x\to 0}(f(x_0)+f(x))=f(x_0)+\lim_{x\to 0}f(x)=f(x_0), $which shows that $f(x)$ is continuous at $x_0$.
