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This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values. The oscillation of a function at a point quantifies these discontinuities as follows: in a removable discontinuity, the distance that the value of the function is off by is the oscillation;
Classification of discontinuities – Wikipedia – Thomae’s function is discontinuous at every rational point, but continuous at every irrational point. The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere.
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