Question

Let $f\left(x\right)$ be a function in $[0,5]$ such that $f^2\left(x\right)=1\forall x\in \left[0,5\right]$ and $f\left(x\right)$ is discontinuous at all integers in $[0,5]$. Find total number of fuctions.

Answer 1

$\begin{array}{c}As{f}^2\left(x\right)=1;f\left(x\right)=1or-1\\ f:[0,5]\to \{-1,1\}\\ f\left(x\right)isdiscontinuousatallintegersin[0,5]\\ f\left(0\right)=-1,f\left(0^{+}\right)=1,f\left(1\right)=1,......\\ Nowateachintegerwehavetwooptions\\ 6times2=12\\ Also,wecanhavef\left(0\right)=1andsoon\\ So,12\times 2=24functions\end{array}$