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I tried considering differentiability of y=arctan(x) with this formula $f'(a)=\lim\limits_{x \to a} \frac{f(x)-f(a)}{x-a}$ for certain points and it was (p.s how to consider differentiability for all points).

On the other hand $$\text{sgn}(x-1)=\begin{cases} 1, & x>1\\0, & x=1\\-1, & x<1 \end{cases}$$ RHL≠LHL implies that $$\lim_{x\to 1 }sgn(x-1) $$ does not exist so the function y=sgn(x-1) is not continous at x=1 that means that is not differentiable at x=1 .

Need a bit help to summarize all of this Calculus I problem .

P.s i mean how to consider main function $y=\arctan(x)+x*sgn(x-1)$ differentiability ?

Thank you in advance :)

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The function is not even continuous at $x=1$. And from contraposition: $$\text{differentiable}\Rightarrow \text{continuous}$$ $$ \text{not continuous}\Rightarrow \text{not differentiable}$$ So the function is not differentiable at the point.

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  • $\begingroup$ Thank you for your answer @Marek but based in the graph I can understand that is not differentiable at x=1 but i need to prove that mathematically :) $\endgroup$ – John Oct 18 '20 at 14:47
  • $\begingroup$ My proof is mathematical. Contraposition is the law of logic. And differentiability implies continuity is a known fact. $\endgroup$ – Marek Kryspin Oct 18 '20 at 15:04
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    $\begingroup$ And if you want to prove the discontinuity, count the left and right limit. $\endgroup$ – Marek Kryspin Oct 18 '20 at 15:06
  • $\begingroup$ Thank you @Marek :) $\endgroup$ – John Oct 18 '20 at 15:24

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