Exercise 13, § 2

Let $a$ be an irrational number $> 0$. For each rational number $x$, let $f_a(x)$ be the real number in the interval $[0, a[$ such that $x - f_a(x)$ is an integral multiple of $a$. Show that $f_a$ is a continuous injective mapping of $\mathbf{Q}$ into $[0, a[$. Deduce with the help of Exercise 9 that there exist continuous bijective mappings of $\mathbf{Q}$ onto itself, whose inverse mapping is not continuous at any point.