A function $f:\mathbb{R^2}->\mathbb{R}$ is called degenerate on $x_i$ if $f(x_1,x_2)$ remains constant when $x_i$ varies ($i=1,2$). Define $$f(x_1,x_2)=\mid 2^{\pi i / x_1}\mid^{x_2}\text{ for }x_1 \neq 0$$ where $i=\sqrt{-1}$. Then which of the following statements is true?
a. $f$ is degenerate on both $x_1$ and $x_2$
b. $f$ is degenerate on $x_1$ but not on $x_2$
c. $f$ is degenerate on $x_2$ but not on $x_1$
d. $f$ is neither degenerate on $x_1$ nor on $x_2$
I think the answer should be $(d)$ but it is given as $(a)$. It was in a test, so I would challenge the answer if I'm correct. Can someone please verify or show why it's $(a)$? Thank you.
Note that $\left | 2^{\pi i/x_i} \right | = 1 $ for all $x_i \in \mathbb{R}, \, x_i \neq 0$. The answer (a) should be clear from that statement.