Is there any other function $f(x)$ such that $f$ is Discontinous at $x=a$ but $fof$ is continous Is there any other function $f(x)$ such that $f$ is Discontinous at $x=a$ but $fof$ is continous.
I know the well know example for this as Dirchlet's function. is there any other?

The function
$$
f(x) = \begin{cases}a - 1 & x\ge a \\ a - 2 & x < a \end{cases}
$$
is discontinuous at $x = a$, but $f(f(x)) = a - 2$ is a continuous (constant) function.
