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Let $f:\mathbb R \to \mathbb R$ be a continuous function in $0$ such that $$f(x+y)=f(x)+f(y) \quad\forall x,y\in \mathbb R$$ Prove that $f$ is continuous on $\mathbb R$.

I'm not asking for the proof because I know this question has already been asked in MSE. What I want is to verify whether my proof is right or wrong.

Clearly $f(0)=0$ and we have $$\lim_{y\to 0}f(x+y)=\lim_{y\to 0 }f(x)+f(y)$$ Note that $g(x)=f(x)+f(a) $ is indeed continuous in $0$ ($a$ here is constant). Hence $$\lim_{y\to 0}f(x+y)=\lim_{y\to 0 }f(x)+f(y)=f(x)$$ Now a small change of variable $u=x+y$ will transform this into $$\lim_{u\to x}f(u)=f(x) \quad \forall x \in \mathbb R$$ Am I right?

PNT
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    Yes, your proof is fine. Perhaps the order of the first two statements may need a little rectification. I'd say this : $f(0)=0$, then $\lim_{y \to 0} f(y) = 0$ by continuity, then $\lim_{y \to 0} f(x)+f(y) = f(x)$ for any $x$, and then $\lim_{y \to 0} f(x+y) = f(x)$ for any $x$ by definition. The third line then follows this logic. – Sarvesh Ravichandran Iyer Oct 13 '22 at 10:23
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    The proof is fine in concept yes, however you shouldn't write "$\lim$" of something when you haven't proved the limit exists. Instead, you should write for example : "$f(x+y) = f(x) + f(y) \xrightarrow[y \to 0]{} f(x) + f(0) = f(x)$ by continuity of $f$ in $0$" or something along those lines, in my opinion – Bruno B Oct 13 '22 at 10:25

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