The function $\text{⨍ (x)}$ is discontinuous only at $\mathrm{x}=0$ such that ${\text{⨍ }}^2\text{(x)=1∀x∈R, ∀x∈R.}$ Then the total number of such function is

$$\begin{gathered} \text{f}\left(x\right)\text{=}\left\{\begin{array}{ll}1& \mathrm{x}\le 0\\ -1& \mathrm{x}>0\end{array}\text{ f}\left(x\right)\text{=}\left\{\begin{array}{ll}-1& \mathrm{x}\le 0\\ 1& \mathrm{x}>0\end{array}\right.\right. \\ \text{f}\left(x\right)\text{=}\left\{\begin{array}{ll}1& \mathrm{x}<0\\ -1& \mathrm{x}\ge 0\end{array}\text{ f}\left(x\right)\text{=}\left\{\begin{array}{ll}-1& \mathrm{x}<0\\ 1& \mathrm{x}\ge 0\end{array}\right.\right. \\ \text{f}\left(x\right)=\left\{\begin{array}{cc}1& \mathrm{x}>0\\ -1& \mathrm{x}=0 \\ 1& \mathrm{x}<0\end{array}\right.\text{ f}\left(x\right)=\left\{\begin{array}{cc}-1& \mathrm{x}>0\\ -1& \mathrm{x}<0\\ 1& \mathrm{x}=0\end{array}\right. \end{gathered}$$

There are six functions