6. The sign function is defined as $\operatorname{sgn} x = \begin{cases} 1, & x > 0, \\ 0, & x = 0, \\ -1, & x < 0. \end{cases}$ Let $f(x)$ be an increasing function on $\mathbf{R}$, and $g(x) = f(x) - f(ax)$ where $a > 1$. Then\\
A. $\operatorname{sgn}[g(x)] = \operatorname{sgn} x$\\
B. $\operatorname{sgn}[g(x)] = -\operatorname{sgn} x$\\
C. $\operatorname{sgn}[g(x)] = \operatorname{sgn}[f(x)]$\\
D. $\operatorname{sgn}[g(x)] = -\operatorname{sgn}[f(x)]$