# Function of bounded variation

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A function whose variation is bounded (see Variation of a function). The concept of a function of bounded variation was introduced by C. Jordan  for functions of one real variable in connection with a generalization of the Dirichlet theorem on the convergence of Fourier series of piecewise-monotone functions (see Jordan criterion for the convergence of a Fourier series). A function defined on an interval is a function of bounded variation if and only if it can be represented in the form , where (resp., ) is an increasing (resp., decreasing) function on (see Jordan decomposition of a function of bounded variation). Every function of bounded variation is bounded and can have at most countably many points of discontinuity, all of the first kind. A function of bounded variation can be represented as the sum of an absolutely-continuous function (see Absolute continuity), a singular function and a jump function (see Lebesgue decomposition of a function of bounded variation).