Question

If $f\left(x\right)=\{\begin{array}{cc}x& \text{if}x\text{is rational}\\ 1-x& \text{if}x\text{is irrational}\end{array}$, then the number of points for $xϵR$ where $f\left(f\right(x\left)\right)$ is/are discontinuous

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

$f\left(f\right(x\left)\right)=\{\begin{array}{cc}f\left(x\right),& f\left(x\right)\text{is rational}\\ 1-f\left(x\right),& f\left(x\right)\text{is irrational}\end{array}$

$=\{\begin{array}{cc}x,& x\text{is}\text{rational}\\ 1-(1-x),& x\text{is}\text{irrational}\end{array}$

So $f\left(f\right(x\left)\right)=x\forall xϵR$
So $f\left(f\right(x\left)\right)\text{is continuous}\forall xϵR$