an idempotent operation is one that has no additional effect if it is called more than once with the same input parameters. ... In mathematics, an idempotent operation is one where $f(f(x)) = f(x)$
Possible functions $= 2^2 = 4$
$f(0)=0, f(1)=0$
$F(0)=0, f(1)=1 $
$F(0)=1, f(1)=0$ // NOT IDEMPOTENT
$f(0)=1, f(1)=1$
if $a=0$
$f(f(a)) = f(a)$
$f(f(0)) = f(0)$
$1.$
$if f(0)=0$
$f(0) = f(0)$ // idempotent
$2.$
if $f(0)=1$
then $f(1)=f(0)$ Here we have assumed that $f(0)=1$ and for this function being NOT idempotent $f(1)$ will be $f(1)=0$
Therefore, (C) is the correct option.