Question

Let $f\left(x\right)=\{\begin{array}{ccc}x& if& xisrational\\ 2-x& if& xisirrational\end{array}$ Then $fof\left(x\right)$ is continuous

• A. everywhere
• B. no where
• C. at all irrational $x$
• D. at all rational $x$

Answer 1

Answer: (A)

everywhere

If $x$ is rational $f\left(f\right(x\left)\right)=f\left(x\right)=x$.
If $x$ is irrational $f\left(f\right(x\left)\right)=f(2-x)=x$.
Now $f\left(f\right(x\left)\right)=x$ for all $x$. Hence this is continuous on all $x$.