Which of the following is always correct

In (A) $f\text{'}\left(x\right)>0$  for every $x\in D_f$

$\cancel{\Rightarrow }f\left(x\right)$  is one-one

ex.  $f\left(x\right)=\tan x$

In (B), $f\text{'}\left(x\right)<0\forall x\in D_f\cancel{\Rightarrow }f$  is one-one

ex.  $f\left(x\right)=\cot x$

In (C), $\left|f\right(x\left)\right|$  is continuous $\cancel{\Rightarrow }f\left(x\right)$  is continuous

ex. $f\left(x\right)$ $=\{\begin{array}{c}1:x<0\\ -1:x\ge 0\end{array}$  is discontinuous at $x=0$

But $\left|f\right(x\left)\right|=1\forall x\in R$  which is continuous

$\therefore$  (A), (B), (C) are not correct

In (D), $f\left(a\right)=2$ , $x=a$  is a point of local minimum and $f\left(x\right)$  is continuous

$\therefore f(a+h)>2,f(a-h)>2$

$\Rightarrow \left[f\right(a+h\left)\right]=\left[f\right(a-h\left)\right]=2$

Hence $\left[f\right(x\left)\right]$  is also continuous at $x=a$

$\therefore$  (D) is correct.