Problem: Suppose $f:\mathbb{R}\to \mathbb{R} $ satisfies $f(x+y)=f(x)+f(y)$ for each $x,y\in \mathbb{R} $. Show that
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$f(nx)=nf(x)$ for all $x\in \mathbb{R} , n\in \mathbb{N}$;
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$f$ is continuous at a single point if and only if $f$ is continuous on $\mathbb{R} $;
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$f$ is continuous if and only if $f(x)=\alpha x$ for some $\alpha \in \mathbb{R} $.