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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...**

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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.

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