Identify the correct statement,where $[.]$ and $\{.\}$ denote the greatest integer function and fractional part function,respectively.

• A
  If $f(x)$ is a differentiable and increasing function,then $g(x) = f(f(x)) + 1$ is a decreasing function.
• B
  If $x \in (0, 1)$,then $[x][\sin x] \neq [x \sin x]$.
• C
  $f(x) = \{\cos x\}\{\cos^2 x\}\{\cos^3 x\}$ is a continuous function in $[0, \frac{\pi}{2}]$.
• D
  $f(x) = \{x\}\{\sin x\} + \{x \sin x\}$ is a differentiable function in $x \in (0, 1)$.