Recall that a real-valued function defined on an interval is (in) Baire class one () iff it is the pointwise limit of continuous functions.
Examples are continuous functions, of course, but functions in do not need to be continuous. An easy example is the function given by if and if . This is the pointwise limit of the functions . By the way, an easy modification of this example shows that any function that is zero except at finitely many points is in .
Step functions are another source of examples. Suppose that and that is constant on each . Then is the pointwise limit of the functions , defined as follows: Fix a decreasing sequence converging to , with and for all . Now define as the restriction of to
and let extend by joining consecutive endpoints of the components of its domain with straight segments.
An important source of additional examples is the class of derivatives. Suppose and for all . This is the pointwise limit of the functions given by
This simple construction does not quite work if is defined on a bounded interval (as may fall outside the interval for some values of ). We can modify this easily by using straight segments as in the case of step functions: Say . For large enough so , define as above for , and now set and extend linearly in the interval .
Additional examples can be obtained by observing, first, that is a real vector space, and second, that it is closed under uniform limits (the latter is not quite obvious). This gives us, for instance, that all monotone functions are in , since monotone functions are the uniform limit of step functions on bounded intervals: Given an increasing , let . It follows that all functions of bounded variation are in , since any such function is the difference of two increasing functions.
Another interesting source of examples is characteristic functions. Given , the function is in iff is both an and a set.
On the other hand, is not in , since it is discontinuous everywhere while Baire class one functions are continuous on a comeager set.