2

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$. If $f$ is continuous at zero how can I prove that is continuous in $\mathbb{R}$.

Martin Sleziak
  • 56,060
Ivan S. Guerra
  • 517
  • By sliding around a neighborhood of zero. In other words if $\delta$ is very close to zero, then by continuity, so is $f(\delta)$, and so is $f(x + \delta) = f(x) + f(\delta)$. Now just formalize that argument. – user4894 Oct 20 '14 at 19:59
  • See also here: http://math.stackexchange.com/questions/93816/proving-that-an-additive-function-f-is-continuous-if-it-is-continuous-at-a-sin (And as was already pointed out, you can find more related posts here: http://math.stackexchange.com/questions/423492/overview-of-basic-facts-about-cauchy-functional-equation) – Martin Sleziak Feb 15 '16 at 13:51

3 Answers3

2
  1. Show $f(0)=0$
  2. For any $x\in\mathbb{R}$, $|f(x+h)-f(x)|=|f(h)|$
Julián Aguirre
  • 77,704
1

Prove it is Lipschitz. (Actually you can prove it is of the form $f(x) = cx$ for some constant $c$.) See also this thread.

Community
  • 1
Tomasz Kania
  • 16,996
1

Hint: Use the definition, prove that $f(0)=0$ and use $$f(a+\varepsilon)-f(a)=f(\varepsilon)\,.$$

Berci
  • 92,013