Which of the following is true 

If $\left|f\left(x\right)\right|$ is continuous on $(a,b)$, then the function $f$ must be continuous on  $(a,b)$

For every continuous functions  $f\left(x\right),g\left(x\right)$ on $(0,\infty )$ satisfying $f\left(x\right)<g\left(x\right)$ for all $x>0$ and both  $\underset{x\to \infty }{\lim }f\left(x\right)and\underset{x\to \infty }{\lim }g\left(x\right)$ exist, then $\underset{x\to \infty }{\lim }f\left(x\right)<\underset{x\to \infty }{\lim }g\left(x\right)$.

If both functions $f\left(x\right)$ and  $g\left(x\right)$ are discontinuous at x=a, then $f\left(x\right)$ $g\left(x\right)$is always discontinuous at x=a

For every continuous monotonic functions  $f\left(x\right),g\left(x\right)$  on $ℝ$, Then their sum $f\left(x\right)+g\left(x\right)$ is need not be monotonic on $ℝ$.