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}