Suppose that $f$ satisfies $f(x+y)=f(x)+f(y)$, and that $f$ is continuous at $0$. Prove that $f$ is continuous at $a$ for all $a$

Since $f$ is continuous at $0$...$$\forall \epsilon>0\exists \delta>0(\forall x\in \mathbb{R})[|x-0|<\delta\Longrightarrow |f(x)-0|]<\epsilon$$

It is quite easy to see that $f(0)=0$ which implies $f(a-a)=f(a)+f(-a)=0$. This implies $f(x-a)=f(x)-f(a)$. If we make the substitution $x-a$ for $x$ we have $$\forall \epsilon>0\exists \delta>0(\forall x\in \mathbb{R})[|(x-a)-0|<\delta\Longrightarrow |f(x-a)-0|]<\epsilon$$ which implies$$\forall \epsilon>0\exists \delta>0(\forall x\in \mathbb{R})[|x-a|<\delta\Longrightarrow |f(x)-f(a)|]<\epsilon$$

Q.E.D

I would like to know if the $x-a$ substitution is valid, and if I have the right idea. Thanks!