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Prove that if $f$ is differentiable at $x_{0}$ then there exists a $ \delta>0 $ and a $K_{0}>0$ such that for all $x\in N_{\delta}(x_{0})$, $|f(x)-f(x_{0})|\leq K_{0}|x-x_{0}|$

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closed as off-topic by Care Bear, Sami Ben Romdhane, martini, user1729, Mathmo123 Jul 30 at 10:00

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When you ask homework questions, please explain what you've tried so far and what in particular is giving you trouble. It's not very useful if we just give you the answer. –  Jonathan Christensen Dec 4 '12 at 16:17
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2 Answers 2

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$$\lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0)\Longrightarrow \,\forall\,\epsilon >0\,\,\exists\,\delta>0\,\,\,s.t.$$

$$|x-x_0|<\delta\Longrightarrow\left|\frac{f(x)-f(x_0)}{x-x_0}-f'(x_0)\right|<\epsilon\Longleftrightarrow $$

$$|f(x)-f(x_0)|<\left(f'(x_0)+\epsilon\right)|x-x_0|\,\,,\,\,\forall x\in N_\delta(x_0):=\{x\in\Bbb R\;;\;|x-x_0|<\delta\}$$

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Note the homework tag. I like this answer on how to handle homework questions. I prefer to leave at most a hint. –  robjohn Dec 4 '12 at 18:38

Hint $$ \lim_{x\to x_0}\frac{f(x)-f(x_0)}{x-x_0}=f'(x_0) $$ Consider the definition of a limit.

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