Function discontinuous everywhere but its composition with itself is continuous

[nietzsche]

Homework Statement

Give an example of a function f(x) that is discontinuous at each point of its domain, whereas f(f(x)) is continuous everywhere.

Homework Equations

The Attempt at a Solution

I guess the function has to be defined piecewise, but I don't know how to approach this at all. I'm thinking of making conditions for x rational, x irrational, but I wouldn't know how to make the function discontinuous at ALL irrational points. Maybe the function doesn't need to take on values at all x? I really don't know.

 

[nietzsche]

never mind, i figured it out