Let $$f:\mathbb{R}\rightarrow\mathbb{R}$$ be a function such that for any $$x,y\in\mathbb{R}$$ $$f(x+y)=f(x)+f(y)$$. Prove/Disprove $$f$$ must be continuous.

Proof We have $$\lim_{x\rightarrow a}f(x)=f(a)$$ for some $$a\in\mathbb{R}.$$ Then, Then, we want to prove $$\lim_{x\rightarrow t}f(x)=f(t)$$ for any $$t\in\mathbb{R}.$$ So

$$\lim_{x\rightarrow t}f(x)=\lim_{x\rightarrow a}f(x-a+t)=\lim_{x\rightarrow a}f(x)-f(a)+f(t)=f(a)-f(a)+f(t)=f(t)$$

for any $$t\in\mathbb{R}.$$

I have a question that: why we have $$\lim_{x\rightarrow a}f(x)=f(a)$$ for some $$a\in\mathbb{R}.$$ How do you know this? Can you explain? Thanks...**

why we have $$\lim_{x\rightarrow a}f(x)=f(a)$$ for some $$a\in\mathbb{R}.$$ How do you know this?
We don't know this, because in general it's not true. There may not exist any such $$a$$. Indeed, assuming the axiom of choice, there do exist discontinuous additive functions; see Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$? However, your argument does show that such a function has to be everywhere discontinuous; an additive function which is continuous at one point must be continuous everywhere.