The class of continuous functions is called the Baire class 0. For each 
, the functions that can be considered as pointwise limits
of sequences of functions of Baire class 
but are not in any of the preceding classes are said to
be of Baire class 
.
A Baire or (analytically representable) function is that function belonging to a
Baire class 
for some 
.
For example, discontinuous functions representable
by Fourier series belong to class 1.
This notion was introduced by Baire in the 19th century. In 1905, Lebesgue showed that each of the Baire classes is nonempty and that there are (Lebesgue-) measurable functions that are not Baire functions (Kleiner 1989).
