The domain of the derivative of the function f(x)=tan−1⁡x     if |x|≤112(|x|−1) if |x|>1

f(x)=11+x2,  if x<1  and is will defined.              f(x)=12,  if x>1              = −12,  if x<−1.Right derivative at x=1 is x→1lim12=12Left derivative at x=1 is x→1lim11+x2=12Hence,  f(x) at x=1 is 12.                   f(x) at x=−1 does not exist∴                f(x) is well defined at all points except −1.Hence (3) is the correct answer.