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Let $D\subseteq \mathbb{R}$ and let $f:D\rightarrow \mathbb{R}$. We say that the function $f$ is an $\mathcal{L}$-function if there exists some constant $K\geq 0$ for which $\left|f(x)-f(y)\right|\leq K\left|x-y\right|$.

1.) Prove that every $\mathcal{L}$-function function is uniformly continuous on its domain.

2.) Give an example showing that there exist uniformly continuous functions which are not $\mathcal{L}$-functions.

3.) Prove that if $f:(a,b)\rightarrow \mathbb{R}$ is an $\mathcal{L}$-function and is differentiable, then $f'$ is bounded.

4.) Prove or disprove that a function is an $\mathcal{L}$-function on $(a,b)$ if and only if it is differentiable on $(a,b)$.

Response: So far I haven't gotten any work worth showing.

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    I believe $f(x)=\sqrt x$ works for part (2).2012-12-10
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    *So far I haven't gotten any work worth showing.* You mean you do not know how to solve (1)? If you spent two minutes thinking about it with a pen and a piece of paper in front of you and if you have the slightest idea of the definitions involved, I find this simply difficult to believe.2012-12-10
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    @did Let me one up you and say I've spent 5 minutes and nothing is exactly popping out.2012-12-10
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    Once again: you pretend you spent 5 minutes thinking about question (1) with the definition of uniform continuity and the hypothesis of the exercise both written on a slip of paper placed in front of you, and that *nothing popped out*?2012-12-10
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    All right, let's try to be polite. did is right though, 1) is completely trivial from the definition of uniform continuity. Write it out in full detail with epsilons and all, and it should be simple. 3) also follows directly from the definition of differentiability. (Perhaps more than 5 minutes on a problem is appropriate before it ends up on stackexchange?)2012-12-10
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    In general, when trying to prove a statement like "If A, then B," you should write down the definitions involved in A, and also the definitions involved in B. Then see if you can play around with these definitions to go from A to B. This helps for both (1) and (3).2012-12-10

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