Functions of Bounded Variation

# Functions of Bounded Variation

 Definition: Let $f$ be a function on the closed interval $[a, b]$. The Variation of $f$ associated with the partition $P$ denoted $V_f (P)$ is defined to be $V_f (P) = \sum_{k=1}^n \mid f(x_k) - f(x_{k-1}) \mid$. The function $f$ is said to be a function of Bounded Variation if there exists a positive real number $M > 0$ such that for all partitions $P \in \mathscr{P} [a, b]$ we have that $\displaystyle{V_f (P) \leq M}$.

We will now look at some nice theorems regarding functions of bounded variation. Recall from the Boundedness Theorem page that if $I = [a, b]$ is a closed and bounded interval and $f : I \to \mathbb{R}$ is a continuous function then $f$ is bounded on $I$. In the subsequent theorem, we will see that if a function $f$ (not necessarily continuous) is of bounded variation on that interval $[a, b]$ then $f$ is also bounded on $[a, b]$.

 Theorem 1: If $f$ is a function of bounded variation on the interval $[a, b]$ then $f$ is bounded on $[a, b]$.

Note that $f$ need not be continuous on $[a, b]$. We already know that a function that is continuous on $[a, b]$ then it is necessarily bounded on $[a, b]$!

• Proof: Let $f$ be a function of bounded variation on the interval $[a, b]$. Then there exists a positive real number $M > 0$ such that for all partitions $P \in \mathcal P [a, b]$ 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}
• For all $x \in [a, b]$ consider the partition $P = \{ a, x, b \}$ (where $P = \{ a, b \}$ if $x = a$ or $x = b$). Then:
(2)
\begin{align} \quad V_f ( \{ a, x, b \}) = \mid f(b) - f(x) \mid + \mid f(x) - f(a) \mid \leq M \end{align}
• Hence we have that $\mid f(x) - f(a) \mid \leq M$, so for all $x \in [a, b]$ we have that $\mid f(x) \mid \leq \mid f(a) \mid + M$, so $f$ is bounded on $[a, b]$. $\blacksquare$ 