# How to consider differentiability of this function $y=\arctan(x)+x*sgn(x-1)$?

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 :)

## 1 Answer

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.

• 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 :) – John Oct 18 '20 at 14:47
• My proof is mathematical. Contraposition is the law of logic. And differentiability implies continuity is a known fact. – Marek Kryspin Oct 18 '20 at 15:04
• And if you want to prove the discontinuity, count the left and right limit. – Marek Kryspin Oct 18 '20 at 15:06
• Thank you @Marek :) – John Oct 18 '20 at 15:24