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Let $f:R\rightarrow R$ be a function such that $f(x+y)=f(x)+f(y)\forall x,y\in R$ if $f(x)$ is differentiable at $x=0$ then

$\begin{array}{1 1}(a)\;f(x)\;is\;differentiable\;only\;in\;a\;finite\;interval\;containing\;zero\\(b)\;f(x)\;is\;continuous\;\forall\;x\in R\\(c)\;f'(x)\;is\;not\;constant\;\forall\;x\in R\\(d)\;f(x)\;is\;differentiable\;except\;at\;finitely\;many\;points\end{array}$

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$f(x+y)=f(x)+f(y)\forall x,y\in R$
Putting $x=y=0$ we get,
Also $f'(x)=\lim\limits_{h\to 0}\large\frac{f(x+h)-f(x)}{h}$
$\Rightarrow \lim\limits_{h\to 0}\large\frac{f(h)}{h}$$=f'(0)=k$ say
$\Rightarrow f(x)=kx+c$
But $f(0)=0\Rightarrow c=0$
$\therefore f(x)=kx$
Which is continuous and differentiable $\forall x\in R$
answered Dec 31, 2013 by sreemathi.v

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