If $f(x)=\large\frac{1}{1-x}$,find the points of discontinuity of the composite function $y=f(f(f(x)))$

Question

$(a)\;0,0\qquad(b)\;0,1\qquad(c)\;1,1\qquad(d)\;None\;of\;these$

Answer 1

$f(x)$ is discontinuous at $x=1$

Now $f(f(x))=\large\frac{1}{1-\Large\frac{1}{1-x}}$

$\Rightarrow \large\frac{x-1}{x}$ which is discontinuous at $x=0$

But $f(f(f(x)))=\large\frac{1}{1-\Large\frac{x-1}{x}}$$=x$

Which has no point of discontinuity.

Hence $x=0,1$ are the only points of discontinuity.

Hence (b) is the correct answer.