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

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SUMMARY

The discussion focuses on constructing a piecewise function f(x) that is discontinuous at every point in its domain while ensuring that f(f(x)) is continuous everywhere. The solution involves defining f(x) based on the rationality of x, specifically by assigning different values for rational and irrational inputs. This approach guarantees that f(x) is discontinuous at all points, yet the composition f(f(x)) results in a continuous function across its domain.

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  • Understanding of piecewise functions
  • Knowledge of rational and irrational numbers
  • Familiarity with continuity and discontinuity in functions
  • Basic concepts of function composition
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  • Study the concept of continuity and discontinuity 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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