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Let $f : \mathbb{R}\rightarrow\mathbb{R}$ be a function with the property $f(x + y) = f(x) + f(y)$ for all $x,y\in \mathbb{R}$. Assume that $\displaystyle\lim_{x\rightarrow 0}f(x) = L$.

1.Calculate $L$.

2.Show that $f$ is continuous at all points in its domain.


marked as duplicate by Adriano, PVAL-inactive, hardmath, Hurkyl, user147263 Nov 19 '14 at 20:15

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1. Consider $f(0+x)$ to get $f(0)$.

Now consider $f(2x)=2f(x)$, which by iterating tells us that $f(\frac{x}{2^n})=\frac{1}{2^n}f(x)$.

2. See that $\displaystyle\lim_{x \rightarrow a} f(x) = \lim_{x \rightarrow x_0} f(x - x_0 + a)$


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