calculus - Proving that $f(x+y) = f(x) + f(y)$ and $f$ being continuous at one point implies that $f$ is continuous on ${bf R}$

Suppose $f(x+y)=f(x) + f(y)$ for all $x,y \in \mathbb R$ and $f$ is continuous at a point $a \in \mathbb R$. Prove that $f$ is continuous at every $b \in \mathbb R$.

I know that in order to prove continuity we can use the definition that states that $\lim_{x\to b}f(x+y)=f(b)$ then the function is continuous however I do not know how to show that the limit will be $f(b)$ for the function. Thanks in Advance.