Function $f\left(x\right)=\left|x\right|-|x-1|$ is monotonically increasing when

 $f\left(x\right)=\left|x\right|-|x-1|$                 

$$\begin{gathered} f\left(x\right)=\{\begin{array}{ll}-x-(1-x),& x<0\\ x-(1-x),& 0\le x<1\\ x-(x-1),& x\ge 0\end{array} \\ f\left(x\right)=\{\begin{array}{lc}-1,& x<0\\ 2x-1,& 0\le x<1\\ 1,& x\ge 1\end{array} \end{gathered}$$

graph of the function shows that $f\left(x\right)$  clearly increases in $(0,1)$