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 $[a, b]$.