Pointwise discontinuous functions
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The concept of pointwise discontinuity is a fairly recent one in mathematics. Originally introduced as a convenient term in the study of integration, it has quite outgrown its former sphere of usefulness and has had an ever-widening field of application in modern analysis. The appearance in 1899 of the doctor's thesis of M. Baire, in which he investigated the properties of a function approached by continuous functions and found that the necessary and sufficient condition for such approach involves the idea of pointwise discontinuity, firmly grounded the conception in the fundamentals of mathematical theory. The considerations of the present paper involve a number of investigations into certain phases of the subject of pointwise discontinuity; such as, the construction and classification of pointwise discontinuous functions; their properties, singly and in combination, etc. We have just remarked that this subject is closely related to the question of approach of continuous functions, and this phase of the subject is treated in Chapter III. The final chapter is devoted to a short study of the oscillation function in the general case.
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