Bachelor of Science

In this academic exercise, we study the oscillation index and the convergence index of Baire-l functions. The main purpose of this project is to supply examples of functions of different oscillation indices, and sequences of continuous functions with different convergence indices. We also study some theorems related to the relationship of the indices, in particular, we supply the details for the proof of Proposition 2.2 in [l], and consider several consequences to this result.

In this academic exercise, we investigate the class of Baire One functions using a recently defined ε - d definition. We give examples and counter-examples of functions which are Baire One, as well as some basic properties including the quasi-uniform convergence of Baire One functions. We collected proofs to show the equivalence of the ε - d definition to earlier definitions of Baire One functions. Then, using the ε - d definition of Baire One functions, we defined a new integral, the Baire One integral. We further developed this new theory with examples and counter-examples of Baire One integrable functions, as well as some basic properties including two kinds of convergence of the Baire One integral. Finally, we established its relationships with the Riemann and Henstock integrals.