Total Variation of a Function

# Total Variation of a Function

Recall from the Functions of Bounded Variation page that a function $f$ is said to be of bounded variation on the interval $[a, b]$ if there exists a positive real number $M > 0$ such that for all partitions $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ we have that:

(1)
\begin{align} \quad V_f (P) = \sum_{k=1}^{n} \mid f(x_k) - f(x_{k-1}) \mid \leq M \end{align}

We will now define the total variation of a function of bounded variation.

 Definition: Let $f$ be a function of bounded variation on the interval $[a, b]$. The Total Variation of $f$ on $[a, b]$ denoted $V_f (a, b)$ is defined to be the least upper bound of the variation of $f$ between all partitions $P \in \mathscr{P}[a, b]$, i.e., $V_f (a, b) = \sup \left \{ V_f (P) : P \in \mathscr{P}[a, b] \right \}$.

We will also often talk about a special function known as a total variation function.

 Definition: Let $f$ be a function of bounded variation on the interval $[a, b]$. The Total Variation Function of $f$ is the function $V : (a, b] \to \mathbb{R}$ defined for all $x \in (a, b]$ by $V(x) = V_f(a, x)$.

Sometimes we will assign $V(a) = 0$ as we'll see later on.