BVfüggvények

BVfüggvények, often translated as "BV functions" or "bounded variation functions," represent a class of functions in real analysis that have a finite total variation. A function $f$ defined on an interval $[a, b]$ is said to have bounded variation if there exists a constant $M$ such that for every partition $P = \{x_0, x_1, \dots, x_n\}$ of $[a, b]$, where $a = x_0 < x_1 < \dots < x_n = b$, the sum of the absolute differences of the function values at the partition points is less than or equal to $M$: $\sum_{i=1}^n |f(x_i) - f(x_{i-1})| \le M$. The smallest such constant $M$ is called the total variation of $f$ on $[a, b]$, denoted by $V_a^b(f)$.

The concept of bounded variation is important in various areas of mathematics, including Fourier analysis, measure theory, and the study of differential equations. Functions of bounded variation possess several useful properties. For instance, they are Riemann integrable, and their derivative exists almost everywhere and is Lebesgue integrable. Moreover, a function is of bounded variation if and only if it can be expressed as the difference of two non-decreasing functions. This decomposition property is a cornerstone in the theory of integration, particularly in the context of the Lebesgue-Stieltjes integral. Examples of BV functions include monotonic functions, continuous piecewise linear functions, and functions whose derivative is absolutely integrable. The set of BV functions forms a vector space.