Consider the following three statements for the function $f : (0,\infty) \rightarrow \mathbb{R}$ defined by
\[
f(x) = \left| \log_e x \right| - |x - 1| :
\]
(I) $f$ is differentiable at all $x>0$.
(II) $f$ is increasing in $(0,1)$.
(III) $f$ is decreasing in $(1,\infty)$.
Then,