Definition:Variation Function

Definition

Let $F : \R \to \R$ be a function that is of bounded variation.

We define the variation function $V_F : \R \to \R$ by:

$\map {V_F} x = \map {V_F} {\hointl {-\infty} x}$

for each $x \in \R$, where $\map {V_F} {\hointl {-\infty} x}$ denotes the total variation of $F$ on $\hointl {-\infty} x$.

Also see

• Results about variation functions can be found here.

Sources

• 2013: Donald L. Cohn: Measure Theory (2nd ed.) ... (previous) ... (next): $4.4$: Functions of Finite Variation