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This post will be essentially about functions of bounded variation of one variable. The main source is the book “*Functions of Bounded variation and Free Discontinuity Problems*” by Ambrosio, Fusco and Pallara. Before we give the definition of a bounded variation function let us recall what exactly does is mean for a function to belong in . Recall that any function can be seen as a *distribution * i.e. as a bounded linear functional on , with

In that case we say that the distribution is representable by the function . Given any distribution we can define its *distributional derivative * to be the distribution defined as

.

In the special case where the distribution can be represented by a function in the way we show above the distributional derivative will be