Showing that a function is continuous for $x$

Question

If $f(x)$ is continuous at the point $x=0$ and for all real numbers $x$ and $y$, the function $f(x+y)= f(x) + f(y)$. Show that $f$ is continuous for all values of $x$.

Not sure where to begin this problem. Any info on it would be appreciated.

Answer 1

For every $x \in \mathbb{R}$, we have $$|f(x +h) -f(x)| = \left|\left(f(x) +f(h)\right) -\left(f(x) +f(0)\right)\right| = |f(h) -f(0)| \underset{h \rightarrow 0}{\longrightarrow} 0$$ since $f$ is continuous at $0$.
This shows that $f$ is continuous on $\mathbb{R}$.

Answer 2

Note that $f(0)=f(0+0)=f(0)+f(0)=2f(0)$

Thus $f(0)=0$

$$\lim_ {h\to 0} f(x+h)-f(x)=\lim_ {h\to 0} f(h)=f(0)=0$$

Therefore $f(x)$ is continuous at $x$