Example of Discontinuous Function: f(x) & f(f(x))

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An example of a function f(x) that is discontinuous at every point of its domain while having f(f(x)) continuous everywhere can be constructed using a piecewise definition. The function can be defined differently for rational and irrational numbers, ensuring it is discontinuous at all points. The challenge lies in ensuring that the composition f(f(x)) results in a continuous function. The discussion highlights the initial confusion about how to approach the problem but concludes with the realization that a solution is possible. This illustrates the complexity of discontinuous functions and their compositions in mathematical analysis.
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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.

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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.
 
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never mind, i figured it out
 

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