# a problem in Lipschitz Functions

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

-
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
Please consider including any working you may have already done - you will find that you get more attention to your question this way. –  Epictetus Dec 4 '12 at 16:17

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

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

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

-
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.

-