Monotonic Functions and Functions of Bounded Variation Review

# Monotonic Functions and Functions of Bounded Variation Review

We will now review some of the recent pages regarding monotonic functions and functions of bounded variation.

• On the Partitions of a Closed Interval page we said that if $I = [a, b]$ is a closed interval then a Partition is a finite set $P = \{ a = x_0, x_1, ..., x_n = b \}$ that satisfies the following inequality:
(1)
\begin{align} \quad a = x_0 < x_1 < ... < x_n = b \end{align}
• We denoted the set of all partitions on $[a, b]$ by $\mathscr{P}[a, b]$. We saw that the partition $P$ divides the interval $[a, b]$ into $n$ subintervals $[x_{k-1}, x_k]$, $k \in \{1, 2, ..., n \}$ whose length is denoted $\Delta x_k = x_k - x_{k-1}$. We also noted that the sum of the lengths of these subintervals is equal to the length of the interval $[a, b]$, i.e.,:
(2)
\begin{align} \quad \sum_{k=1}^{n} \Delta x_k = b - a \end{align}
• On the Monotonic Functions page we said that a function $f$ is Monotonic on $[a, b]$ if it is either increasing or decreasing. We said that a function is Increasing on $[a, b]$ if for all $x, y \in [a, b]$ with $x \leq y$ we have that $f(x) \leq f(y)$, and similarly, $f$ is Decreasing on $[a, b]$ if for all $x, y \in [a, b]$ with $x \leq y$ we have that $f(x) \geq f(y)$.
• If $f$ is an increasing function, $x < y < z$, and $f(y^-) = \lim_{a \to y^-} f(a)$ (the lefthand limit of $f$ at $c$) and $f(y^+) = \lim_{a \to y^+}$ (the righthand limit of $f$ at $c$) then the following inequality always holds:
(3)
\begin{align} \quad f(x) \leq f(y^-) \leq f(y) \leq f(y^+) \leq f(z) \end{align}
• Similarly, if $f$ is a decreasing function and $x < y < z$ then:
(4)
\begin{align} \quad f(x) \geq f(y^-) \geq f(y) \geq f(y^+) \geq f(z) \end{align}
• On the Countable Discontinuities of Monotonic Functions we looked at an important lemma which says that if $f$ is an increasing function on $[a, b]$ and $P = \{ a = x_0, x_1, ..., x_n = b \}\in \mathscr{P}[a, b]$ then the sum of the "jumps" of $f$ at $x_1, x_2, ..., x_{n-1}$ is less than the sum of the jump of $f$ at $a$ to $b$, i.e.,:
(5)
\begin{align} \quad \sum_{k=1}^{n-1} [f(x^+) - f(x^-)] \leq f(b) - f(a) \end{align}
• We then went on to prove that if $f$ is monotonic on $[a, b]$ then $f$ has at most countably infinite many discontinuities on $[a, b]$.
• Then on the Functions of Bounded Variation we said that a function $f$ is of Bounded Variation on the interval $[a, b]$ if there exists an $M \in \mathbb{R}$, $M > 0$ such that for every partition $P \in \mathscr{P}[a, b]$ we have that:
(6)
\begin{align} \quad V_f(P) = \sum_{k=1}^{n} \mid f(x_k) - f(x_{k-1}) \mid \leq M \end{align}
• The value $V_f(P)$ is called the Variation of $f$ associated to $P$.
• We also saw a nice result that showed that if $f$ (not necessarily continuous) is of bounded variation on $[a, b]$ then $f$ is also bounded on $[a, b]$.
• On the Total Variation of a Function page we saw that if $f$ is a function of bounded variation that then the Total Variation of $f$ on $[a, b]$ denoted $V_f (a, b)$ was said to be equal to the supremum of the variation of $f$ associated will all partitions $P \in \mathscr{P}[a, b]$, that is:
(7)
\begin{align} \quad V_f (a, b) = \sup \{ V_f (P) : P \in \mathscr{P}[a, b] \} \end{align}
• We then looked at some properties of the total variation of a function on the Additivity of the Total Variation of a Function page. We first looked at a property regarding the variation of a function $f$. We saw that if $P = \{ a = x_0, x_1, ..., x_n = b \}$ and $\hat{P} = \{ a = x_0, x_1, ..., x_{k-1}, c, x_k, ..., x_n = b \}$, i.e., $\hat{P}$ is $P$ with an additional point $c$ added, then the variation of $f$ associated to $P$ is less than the variation of $f$ associated to $\hat{P}$:
(8)
\begin{align} \quad V_f (P) \leq V_f (\hat{P}) \end{align}
• We used this to prove the additivity of the total variation of a function of bounded variation. If $f$ is of bounded variation on $[a, b]$ and $c \in (a, b)$ then $[a, b] = [a, c] \cup [c, b]$ and the total variation of $f$ on $[a, b]$ equals the total variation of $f$ on $[a, c]$ plus the total variation of $f$ on $[c, b]$:
(9)
\begin{align} \quad V_f (a, b) = V_f(a, c) + V_f(c, b) \end{align}
(10)
\begin{align} \quad V(x) = V_f(a, x) \end{align}
• We proved in a lemma that $V$ is an increasing function. We also proved that the function $V - f$ is an increasing function. Hence, we were able to show that every function $f$ of bounded variation could be written as the difference of two monotonically increasing functions:
(11)
\begin{align} \quad f(x) = V(x) - [V(x) - f(x)] \end{align}