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Let $f$ be a function defined on $R$ such that $$f(x+y) = f(x)+f(y), x, y \in R$$ If $f$ is differentiable at one point of $R$, then prove that $f$ is differentiable on entire $R$.

Here $R$ is Real Numbers Set.

I am doing calculus course in graduation.

Please give me some hints on how to solve this

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1 Answer 1

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Hint:

Let the function be differentiable at $x = a \in \mathbb{R}$. Then $$f'(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}=\lim_{h \to 0 } \frac{f(h)}{h}$$ exists.

Given the above fact, check whether $f'(x)$ exists or not.

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  • $\begingroup$ That suffices. Clear and concise. (+1) $\endgroup$
    – Mark Viola
    Commented Oct 20, 2017 at 15:07

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