​$$\text{Define } f : \mathbb{C} \to \mathbb{C} \text{ by } f(z) = ||z|^2 - 1|^2. \\\text{ Which of the following statements is true?}$$​​

Correct option is D

​$$\textbf{CR Equations Using } \overline{z}: \\[10pt]\text{For a complex-valued function } f(z) = u(x, y) + i v(x, y), \text{ the CR equations are satisfied if:} \\[10pt]\frac{\partial f}{\partial \overline{z}} = 0.$$

Solution :

​$$f : \mathbb{C} \to \mathbb{C} \\[10pt]f(z) =\big| |z|^2 - 1\big|^2\\[10pt]= (z \cdot \overline{z})^2 + 1^2 - 2(z \cdot \overline{z}). \\[15pt]\text{UsingC.R. equation:} \\[10pt]\frac{\partial f}{\partial \overline{z}} = 2z (\overline{z}) - 2z \\[10pt]= 2z (\overline{z} - 1) = 0. \\[15pt]\implies z=0\ or\ \overline{z}=1\\[10pt]\text{so C-R equations are satisfied only at z = 0 and z = 1 . }$$ 

Which means it can only be complex differentiable at $$z\in \{0,1\}$$ .​

​​

​$$\textbf{Hence, Option D is correct}$$​​