If f(x) is defined $\forall x\in R$ and is discontinuous only at x = 0 such that $f^3\left(x\right)-6f^2\left(x\right)+11f\left(x\right)-3=3\forall x\in R,$

then the number of such functions is equal to : 

$\begin{array}{l}\therefore f^3\left(x\right)-6f^2\left(x\right)+11f\left(x\right)-6=0\\ \Rightarrow (f^2\left(x\right)-3f\left(x\right)+2)(f\left(x\right)-3)=0\end{array}$

$\begin{array}{l}\Rightarrow (f\left(x\right)-1)(f\left(x\right)-2)(f\left(x\right)-3)=0\\ \Rightarrow f\left(x\right)=1or2or3\end{array}$   

i) If L.H.L $\ne$ R.H.L $\ne$ f(0) 

Then number of such functions = 3! = 6. 

ii) If only two of 

$L.H.L.f\left(0^{-}\right),R.H.L.f\left(0^{+}\right)andf\left(0\right)$  are equal and third one is no equal then number of such 

functions = ${ }^3{C}_2{\times }^3{C}_2{\times }^2{C}_1=18.$

$\therefore$ Number  of such functions = 24