Let the function $$g : (-\infty, \infty)\rightarrow \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$$ be given by $$g(u) = 2 \tan^{-1}(e^u) - \frac{\pi}{2}$$. Then, g is

A

even and is strictly increasing in $$(0, \infty)$$

B

odd and is strictly decreasing in $$(-\infty, \infty)$$

C

odd and is strictly increasing in $$(-\infty, \infty)$$

D

neither even nor odd, but is strictly increasing in $$(-\infty, \infty)$$