Let $f :\mathbb{R} \rightarrow \mathbb{R}$ be a group homomorphism W. R. T. Usual addition such that $f$ is bounded at a neighborhood of $0$. Can you conclude that $f$ is continuous at $0$?
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3Yeah you can, this is a well known exercise (You can find it in Bartle, Real Analysis, or for a solution in Conjecture and Proof by Laczkovitz) – miraunpajaro Jun 28 '19 at 10:41
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Can you please give me any hint?? – Pradip Jun 28 '19 at 10:57
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Take a look here: https://en.wikipedia.org/wiki/Cauchy%27s_functional_equation Basically once such an $f$ is bounded over any interval, it must be of the form $f(x)=cx$. – Ingix Jun 28 '19 at 12:05
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$\LaTeX{}$ help. – Jul 01 '19 at 05:56
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Also on this site you can find Overview of basic facts about Cauchy functional equation. For example, this might be helpful: Graph of discontinuous linear function is dense. – Martin Sleziak Jul 01 '19 at 10:48