The function $f\left(x\right)=\left|2\mathrm{sgn}\left(2x\right)\right|+2$ has

$f\left(x\right)=\left|2\mathrm{sgn}\left(2x\right)\right|+2$

$=\left|2.\frac{2x}{\left|2x\right|}\right|+2=\left|2.\frac{x}{\left|x\right|}\right|+2$

$f\left(0\right)=2,f\left(0^{+}\right)=|2\times 1|+2=4$

$f\left(0^{-}\right)=\left|2(-1)\right|+2=4$

$f\left(0^{+}\right)=f\left(0^{-}\right)\ne f\left(0\right)$

$\therefore f\left(x\right)$ has removable discontinuity