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a function that is discontinuous at some points. In the functions usually encountered in mathematics, points of discontinuity are isolated, but there exist functions that are discontinuous at all points. An example is the Dirichlet function: f(x) = 0 if x is rational and f(x) = 1 if x is irrational.
The limit of a sequence of continuous functions that converges everywhere may be a discontinuous function. Such discontinuous functions are called functions of the first Baire class, after the French mathematician R. Baire, who provided a classification of discontinuous functions. Measurable discontinuous functions are an important class of discontinuous functions.
H. Lebesgue constructed a theory of the integration of discontinuous functions. N. N. Luzin showed that by changing the values of a measurable function on a set of arbitrarily small measure the function can be made continuous. If a function is monotonic, then it has only jump discontinuities. For functions of several variables, not only isolated points of discontinuity but also, for example, lines and surfaces of discontinuity must be considered.