A Baire class one function is a pointwise limit of continuous functions. On a complete metric domain, its set of discontinuities is meagre; in particular, it has a point of continuity on every nonempty perfect closed subset.
The indicator of the rational numbers in the real numbers is discontinuous everywhere because both the rationals and irrationals are dense. The discontinuity theorem for Baire class one functions therefore shows that it is not Baire class one.

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  1. Baire function
  2. Baire space
  3. Baire category theorem
  4. Complete metric space
  5. Metric space
  6. Topological analysis
  7. Analysis
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