Positive and Negative Variations of a Function

# Positive and Negative Variations of a Function

Recall from the Total Variation of a Function page that if $f$ is a function of bounded variation on $[a, b]$ then the total variation of $f$ on $[a, b]$ is defined to be:

(1)
\begin{align} \quad V_f(a, b) = \sup \{V_f(P) : P \in \mathscr{P}[a, b] \} \end{align}

We will look at two similar concepts known as the total positive variation and the total negative variation of a function of bounded variation.

 Definition: Let $f$ be a function of bounded variation on the interval $[a, b]$ and let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$. Furthermore, let $A(P) = \{ k \in \{0, 1, ..., n \} : f(x_k) - f(x_{k-1}) > 0 \}$ and let $B(P) = \{ k \in \{0, 1, ..., n \} : f(x_k) - f(x_{k-1}) < 0 \}$. The Positive Variation of $f$ associated with $P$ is denoted $p_f(P)$ and is defined to be $\displaystyle{p_f (P) = \sum_{k \in A(P)} \mid f(x_k) - f(x_{k-1}) \mid}$ and the Negative Variation of $f$ associated with $P$ is denoted $n_f(P)$ and is defined to be $\displaystyle{n_f(P) = \sum_{k \in B(P)} \mid f(x_k) - f(x_{k-1}) \mid}$.
 Definition: Let $f$ be a function of bounded variation on the interval $[a, b]$. The Total Positive Variation of $f$ on $[a, b]$ is denoted $p_f(a, b)$ and is defined to be $\displaystyle{p_f(a, b) = \sup \{ p_f (P) : P \in \mathscr{P}[a, b] \}}$ and the Total Negative Variation of $f$ on $[a, b]$ is denoted $n_f(a, b)$ and is defined to be $\displaystyle{n_f(a, b) = \sup \{ n_f(P) : P \in \mathscr{P}[a, b] \}}$.

Before we look at some nice theorems regarding the positive and negative variations of a function, we will first denote some functions. Recall that if $f$ is of bounded variation on $[a, b]$ then the total variation function of $f$ is given by:

(2)
\begin{align} \quad V(x) = V_f (a, x) \end{align}

Similarly we can define the total positive variation function of $f$ to be given by:

(3)
\begin{align} \quad p(x) = p_f (a, x) \end{align}

As well as the total negative variation function of $f$ to be given by:

(4)
\begin{align} \quad n(x) = n_f (a, x) \end{align}

We are now ready to look at some theorems regarding positive and negative variation of a function of bounded variation on the interval $[a, b]$.

 Theorem 1: Let $f$ be a function of bounded variation on $[a, b]$. Then for each partition $P \in \mathscr{P}[a, b]$ we have that $V_f(P) = p_f(P) + n_f(P)$.
• Proof: Let $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$. Then $\Delta f_k = f(x_k) - f(x_{k-1})$ is either positive, negative, or $0$ for each $k \in \{0, 1, ..., n \}$. If $\Delta f_k > 0$ then $k \in A(P)$. If $\Delta f_k < 0$ then $k \in B(P)$. If $\Delta f_k = 0$ then the term $\mid \Delta f_k \mid = 0$ contributes nothing to the sum $V_f(P)$. Therefore:
(5)
\begin{align} \quad V_f(P) = \sum_{k \in A(P)} \mid f(x_k) - f(x_{k-1}) \mid + \sum_{k \in B(P)} \mid f(x_k) - f(x_{k-1}) \mid = p_f(P) + n_f(P) \quad \blacksquare \end{align}
 Theorem 2: Let $f$ be a function of bounded variation on $[a, b]$. Then for all $x \in (a, b]$ we have that $V(x) = p(x) + n(x)$.
• Applying Theorem 1 at the third line and we get:
(6)
\begin{align} \quad V(x) &= V_f(a, x) \\ \quad &= \sup \{ V_f(P) : P \in \mathscr{P}[a, x] \} \\ \quad &= \sup \{ p_f(P) + n_f(P) : P \in \mathscr{P}[a, x] \} \\ \quad &= \sup \{ p_f(P) : P \in \mathscr{P}[a, x] \} + \sup \{ n_f(P) : P \in \mathscr{P}[a, x] \} \\ \quad &= p(x) + n(x) \quad \blacksquare \end{align}
 Theorem 3: Let $f$ be a function of bounded variation on $[a, b]$. If we define $p(a) = 0$ and $n(a) = 0$ then $p$ and $n$ are both nonnegative functions on $[a, b]$.
• Proof: For any partition $P \in \mathscr{P}[a, b]$ we have that:
(7)
\begin{align} \quad p_f(P) = \sum_{k \in A(P)} \mid f(x_k) - f(x_{k-1}) \mid \end{align}
• But each term in the sum is positive, and so $\{ p_f(P) : P \in \mathscr{P}[a, b] \}$ is a nonnegative set of real numbers and so $p(x) = \sup \{ p_f(P) : P \in \mathscr{P}[a, x] \} \geq 0$, so $p$ is a nonnegative function.
• A similar argument holds to show that $n$ is also a nonnegative function. $\blacksquare$