real analysis - $f$ is linear and continuous at a point $implies f$ should be $f(x) =ax,$ for some $a in mathbb R$

Let $f$ be a real valued function defined on $\mathbb R$ such that $f(x+y)=f(x)+f(y)$.
Suppose there exists at least an element $x_0 \in \mathbb R$ such that $f$ is continuous at $x.$ Then prove that $f(x)=ax,\ \text{for some}\ x \in \mathbb R.$

Hints will be appreciated.