Monotonic Functions as Functions of Bounded Variation

# Monotonic Functions as Functions of Bounded Variation

Recall from the Functions of Bounded Variation page that if $f$ is a function on the interval $[a, b]$ and $P = \{ a = x_0, x_1, ..., x_n = b \} \in \mathscr{P}[a, b]$ then the variation of $f$ associated with $P$ is defined to be:

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

Furthermore, $f$ is said to be of bounded variation on $[a, b]$ if there exists a positive real number $M > 0$ such that for all partitions $P \in \mathscr{P}[a, b]$ we have that:

(2)
\begin{align} \quad V_f (P) \leq M \end{align}

We will now show that if $f$ is monotonic on $[a, b]$ then $f$ is of bounded variation on $[a, b]$.

 Theorem 1: If $f$ is a monotonic function on the interval $[a, b]$ then $f$ is of bounded variation on $[a, b]$.
• Proof: Let $P \in \mathscr{P} [a, b]$ where $P = \{ x_0, x_1, ..., x_n \}$. First consider the case when $f$ is increasing. Then the variation of $f$ associated to the partition $P$ is:
(3)
\begin{align} \quad V_f (P) = \sum_{k=1}^{n} \mid f(x_k) - f(x_{k-1}) \mid \end{align}
• Since $f$ is increasing on $[a, b]$ and $x_{k-1} < x_k$ for all $k \in \{ 1, 2, ..., n \}$ we have that $f(x_{k-1}) \leq f(x_{k})$ and so $f(x_k) - f(x_{k-1}) \geq 0$ so $\mid f(x_k) - f(x_{k-1}) \mid = f(x_k) - f(x_{k-1})$ and:
(4)
\begin{align} \quad V_f (P) = \sum_{k=1}^{n} [f(x_k) - f(x_{k-1})] = f(b) - f(a) \end{align}
• Let $M = f(b) - f(a) > 0$. Then for all partitions $P \in \mathscr{P} [a, b]$ there exists an $M > 0$ such that $V_f (P) \leq M$ so $f$ is a function of bounded variation on the interval $[a, b]$. $\blacksquare$