Functions of Bounded Variation on Closed Intervals

This page is intended to be a part of the Measure Theory section of Math Online. Similar topics can also be found in the Real Analysis section of the site.

# Functions of Bounded Variation on Closed Intervals

 Definition: Let $f : [a, b] \to \mathbb{R}$ and let $P = \{ a = a_0, a_1, ..., a_n = b \}$ be a partition of $[a, b]$. Then the Variation of $f$ associated with $P$ is defined as $\displaystyle{V(P, f) = \sum_{i=1}^{n} |f(a_i) - f(a_{i-1})|}$. The function $f$ is said to be of Bounded Variation on $[a, b]$ if there exists an $M \in \mathbb{R}$, $M > 0$ such that for all partitions $P \in \mathscr{P} [a, b]$ we have that $V(P, f) \leq M$.

Here, $\mathscr{P}[a, b]$ denotes the set of all partitions on $[a, b]$.

If $P, P' \in \mathscr{P}[a, b]$ and $P \subseteq P'$, that is, $P'$ is a refinement of $P$, then it is easy to show that:

(1)
\begin{align} \quad V(P, f) \leq V(P', f) \end{align}
 Theorem 1: If $f : [a, b] \to \mathbb{R}$ is of bounded variation on $[a, b]$ then $f$ is bounded on $[a, b]$.
• Proof: Let $x \in [a, b]$ and consider the partition $P = \{ a, x, b \}$. Since $f$ is of bounded variation on $[a, b]$ there exists an $M \in \mathbb{R}$, $M > 0$ such that:
(2)
\begin{align} \quad V(P, f) = | f(x) - f(a) | + | f(b) - f(x) | \leq M \end{align}
• Therefore we have that $| f(x) - f(a) | \leq M$. So $f(x) - f(a) \leq M$. Thus $f(x) \leq M + f(a)$. So $|f(x)| \leq M + |f(a)|$ which shows that $f$ is bounded on $[a, b]$. $\blacksquare$
 Theorem 2: If $f : [a, b] \to \mathbb{R}$ is a monotonic function on $[a, b]$ then $f$ is of bounded variation on $[a, b]$.
• Proof: Let $f$ be monotonically increasing on $[a, b]$ and let $P = \{ a = a_0, a_1, ..., a_n = b \} \in \mathscr{P}[a, b]$ and consider the variation of $f$ associated with $P$:
(3)
\begin{align} \quad V_f(P) &= \sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| \\ &= \sum_{i=1}^{n} (f(x_i) - f(x_{i-1}) \\ &= (f(x_1) - f(x_0)) + (f(x_2) - f(x_1)) + ... + (f(x_n) - f(x_{n-1})) \\ &= f(x_n) - f(x_0) \\ &= f(b) - f(a) \end{align}
• Let $M = f(b) - f(a)$. Then $V_f(P) \leq M$ for all $P \in \mathscr{P}[a, b]$ so $f$ is of bounded variation on $[a, b]$.
• Similarly, if $f$ is monotonically decreasing on $[a, b]$, then:
(4)
\begin{align} \quad V_f(P) &= \sum_{i=1}^{n} |f(x_i) - f(x_{i-1})| \\ &= \sum_{i=1}^{n} (f(x_{i-1}) - f(x_i)) \\ &= (f(x_0) - f(x_1)) + (f(x_1) - f(x_2)) + ... + (f(x_{n-1}) + f(x_n)) \\ &= f(x_0) - f(x_n) \\ &= f(a) - f(b) \end{align}
• Let $M = f(a) - f(b)$. Then $V_f(P) \leq M$ for all $P \in \mathscr{P}[a, b]$ so $f$ is of bounded variation on $[a, b]$. $\blacksquare$

There are many other properties of functions of bounded variation on $[a, b]$. Many of these properties are stated and proven in the Real Analysis hub. We state one result of particular importance:

 Theorem 3: If $f$ is of bounded variation on $[a, b]$ then $f$ can be expressed as a difference of two increasing functions on $[a, b]$, i.e., $f = f_1 - f_2$ where $f_1$ and $f_2$ are increasing on $[a, b]$.