Baire function

Summary

Baire functions are the functions obtained from continuous functions by taking pointwise limits of sequences, again and again. They are sorted into Baire classes indexed by the countable ordinals. Class $0$ is the continuous functions, and class $1$ is the pointwise limits of continuous functions. On $\R$, the Baire functions are exactly the Borel measurable functions. The classes give a fine measure of how far a function is from being continuous.