$$\begin{gathered} f\left(x\right)=2|\mathrm{sgn}(2x)+2| \\ = \left\{\begin{array}{l}4\\ 2\\ 0\end{array}\right.\begin{array}{ll},& x>0\\ ,& x=0\\ ,& x<0\end{array} \end{gathered}$$

At x=0, signum function takes value 0. Also, the modulus ensures positive value of signum function over all other domain. Hence, it is a jump discontinuity which is finite, non-removable,