The function $f\left(x\right)=1+\left[\cos x\right]x$, in $0<x\le \frac{\pi }{2}$

Given : The function $f\left(x\right)=1+\left[\cos x\right]x$, in $0<x\le \frac{\pi }{2}$

To Find : appropriate options with respect to given function and range.

Use given range to find minimum and maximum term of it.

$f\left(x\right)=1$ in $0<x\le \frac{\pi }{2}$

In $0<x\le \frac{\pi }{2},\cos x\in [0,1]$  as $\left[\cos x\right]=0$

Use function $f\left(x\right)$ to find value of $x$,

$f\left(x\right)=10$

$x=1$

$f\left(x\right)$ is a constant function and hence continuous in $\left[0,\frac{\pi }{2}\right]$. It neither strictly increasing nor decreasing.

Therefore, the correct answer is option $\left(3\right)$.