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I'm trying to prove that

The function $f(x)=\tan^{-1}(x)$ is not uniformly continuous on $\Bbb{R}$.

Here's what I've done:

Let $\epsilon>0$ be given.

Now, $$|f(x)-f(y)|=|\tan^{-1}x-\tan^{-1}y|=\Big|\tan^{-1}\left( \frac{x-y}{1+xy}\right) \Big|$$

For non-negative $x,y\in \Bbb{R},$

$$|f(x)-f(y)|=|\tan^{-1}x-\tan^{-1}y|=\Big|\tan^{-1}\left( \frac{x-y}{1+xy}\right) \Big|\leq |\tan^{-1}(x-y)|\leq|x-y|<\delta$$

We can choose $\epsilon=\delta$. Now, for negative $x,y\in \Bbb{R}$. Please, how do I go about it?

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    $\begingroup$ I am afraid that any attempt to prove that claim is doomed to fail. $\endgroup$ – Hagen von Eitzen May 14 '18 at 12:15
  • $\begingroup$ @ Hagen von Eitzen: Then, what should I do? $\endgroup$ – Omojola Micheal May 14 '18 at 12:15
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    $\begingroup$ What you have started on seems to be to prove that it is uniformly continuous... Continue with that! $\endgroup$ – Winther May 14 '18 at 12:16
  • $\begingroup$ Actually, this function is uniformly continuous because of the limits $\lim\limits_{x\to +\infty}f(x)=\frac{\pi}{2}$ and $\lim\limits_{x\to -\infty}f(x)=\frac{-\pi}{2}$. $\endgroup$ – Riemann May 14 '18 at 12:37
  • $\begingroup$ @ Piquito I did not say “continuous and bounded implies uniform continuous.” I mean the existence of limits implies uniform continuous!!!!! OK??? $\endgroup$ – Riemann May 14 '18 at 13:20
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The function $\tan^{-1}x$ is uniformly continuous on $\Bbb{R}$.

By the Mean Value Theorem, for any $x,y\in\Bbb{R}$, $$|\tan^{-1}x-\tan^{-1}y|\leq |x-y|.$$ So for any $\epsilon>0$, take $\delta=\epsilon>0$, when $|x-y|<\delta$, we have $$|\tan^{-1}x-\tan^{-1}y|\leq |x-y|<\delta=\epsilon. $$

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  • $\begingroup$ Your first inequality it would be enough. Regards. $\endgroup$ – Piquito May 14 '18 at 13:07
  • $\begingroup$ Yes, you are right. There is no need to say any word! $\endgroup$ – Riemann May 14 '18 at 13:15

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