= Additive function continuous at one point is linear
{title2=$f(x)=xf(1)$}
A real <additive function> $f:\mathbb R\to\mathbb R$ that is <continuous> at one point $x_0$ is <continuous> everywhere and has form $f(x)=xf(1)$. Indeed $f(0)=0$, and $f(h)=f(x_0+h)-f(x_0)\to0$ as $h\to0$. Then $f(x+h)-f(x)=f(h)\to0$ for every $x$. Additivity gives $f(q)=qf(1)$ for every <rational number>; continuity and the <density of the rational numbers> extend this identity to every real argument.
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