Conclusion
Bounded Variation Functions

The theory of functions with bounded variations developed strongly in functional analysis in the 1990s, driven in particular by its growing use in mathematical imaging and its many practical applications, and more broadly in mathematical engineering.

Spaces of functions with bounded variations are positioned "finely" between spaces of Sobolev functions that are too restricted (not enough functions, especially not "jump" functions) and spaces of Lebesgue functions that are too vast (too many functions). These functions are ideal for dealing with discontinuities.