Function of Bounded Variation
Introduction
In this chapter, we would learn about monotonic functions and their various properties in bounded sets. We shall find that these functions are intimately connected with curves having finite arc lengths.
Properties of Monotonic Functions
Theorem 1 :
Let $f$ be an increasing function defined on $[a, b]$ and let $x0, x1, … , xn$ be $n + 1$ points such that x.
Then we have the inequality,
$k=1$$Σ$$n-1$$[f(x$$k$$+) – f(x$$k$$-)]$ $≤$ $f(b)$ – $f(a)$.
$Proof$: Assume that $y$$k$ $∈$ $(x$$k$$,$ $x$$k+1$$)$. For $1 < k < n – 1$, we have $f(x$$k$$+)$ $≤$ $f(y$$k$$)$ and $f(y$$k-1$$)$ $≤$ $f(x$$k$$-)$, so that $f(x$$k$$+)$ $-$ $f(x$$k$$-)$ $≤$ $f(y$$k$$)$ $-$ $f(y$$k-1$$)$. If we add these inequalities, the sum on the right telescopes to $f(y$$n – 1$$)$ $-$ $f(y$$0$$)$.
Since $f(y$$n – 1$$)$ $-$ $f(y$$0$$)$ $≤$ $f(b)$ $-$ $f(a)$, hence the theorem is proved.
Theorem 2 :
If $f$ is monotonic on $[a, b]$, then the set of discontinuities of $f$ is countable.
$Proof$: Assume that $f$ is increasing and let $S$. be the set of points in $(a, b)$ at which the jump off exceeds $1/m, m > 0$. If $x$$1$$ < x$$2$$ <… <x$$n-1$ are in $S$$m$,
$(n-1)/m$ $≤$ $f(b)$ $-$ $f(a)$.
This means that $S$$m$ must be a finite set. But the set of discontinuities off in $(a, b)$ is a subset of the union $m=1$$∪$$∞$ $S$$m$ and hence is countable.
Theorem 3 :
Let $f$ be strictly increasing and continuous on a compact interval $[a, b]$. Then $f$$-1$ is continuous and strictly increasing on the interval $[f(a), f(b)]$.
Theorem 4 :
Let $I = (a, b)$ be an interval. Let $f : I → ℝ$ be monotonically increasing on $I$. Then at any point $c ∈I$,
$i)$ $f(c – 0)$ $=$ $x∈(a, c)$$sup f(x)$
$ii)$ $f(c + 0)$ $=$ $x∈(c, b)$$inf f(x)$
$iii)$ $f(c – 0)$ $≤$ $f(c)$ $≤$ $f(c + 0)$
Intermediate Value Property
A function $f : [a, b]→ ℝ$ is said to have the intermediate value property on $[a, b]$ if for every $x$$1$, $x$$2$ satisfying $a$ $≤$ $x$$1$ $≤$ $x$$2$ $≤$ $b$ and for every $k$ between $f(x$$1$$)$ and $f(x$$2$$)$, there exists a point $c ∈ (x$$1$$, x$$2$$)$ such that $f(c)=k$.
Note: If a function $f : [a, b]→ ℝ$ be continuous on $[a, b]$, then $f$ has the IVP on $[a, b]$. But if $f : [a, b]→ ℝ$ has the the IVP on $[a, b]$, $f$ need not be continuous on $[a, b]$.
Function of Bounded Variation
Definition 1-
If $[a, b]$ is a compact interval, a set of points
$P = {x0, x1, … , xn}$,
satisfying the inequalities
$a=x0 <x1 <x2 <…<xn=b$,
is called Partition of $[a, b]$. The interval $[(x$$k-1$$) – (x$$k$$)]$ is called the $k$th subinterval of $P$ and we write $Δx$$k$ $=$ $[(x$$k$$) – (x$$k-1$$)]$, so that $k=1$$Σ$$n$$Δx$$k$ $=$ $b – a$.
The collection of all possible partitions of $[a, b]$ will be denoted by $P[a, b]$.
Definition 2-
Let $f$ be defined on $[a, b]$. If $P = {x0, x1, … , xn}$ is a partition defined of $[a, b]$, write $Δf$$k$ $=$ $[f(x$$k$$) – f(x$$k-1$$)]$ for $k = 1, 2, … , n$. If there exists a positive number $M$ such that,
$k=1$$Σ$$n$$|Δf$$k$$|$ $≤$ $M$
for all partitions of $[a, b]$, then $f$ is said to be of Bounded Variation on $[a, b]$.
Theorem 5 :
If $f$ is monotonic on $[a, b]$, then $f$ is of bounded variation on $[a, b]$.
$Proof$: Let $f$ be increasing. Then for every partition of $[a, b]$ we have $Δf$$k$ $≥$ 0 and hence
$k=1$$Σ$$n$$|Δf$$k$$|$ $=$ $k=1$$Σ$$n$$Δf$$k$ $=$ $f(b)$ $-$ $f(a)$.
Theorem 6 :
If $f$ is continuous on $[a, b]$ and if $f’$ exists and is bounded in the interior, say $|f'(x)|$ $≤$ $A$ for all $x$ in $(a, b)$, then $f$ is of bounded variation on $[a, b]$.
$Proof$: Applying Mean Value Theorem, we have
$Δf$$k$ $=$ $f(x$$k$$)$ $-$ $f(x$$k-1$$)$ $=$ $f'(t$$k$$)$$(x$$k$$ – x$$k-1$$)$, where $t$$k$$ ∈ (x$$k-1$$, x$$k$$)$.
This implies,
$k=1$$Σ$$n$$|Δf$$k$$|$ $=$ $k=1$$Σ$$n$$|Δf'(t$$k$$)|$$Δx$$k$ $≤$ $A$$k=1$$Σ$$n$$Δx$$k$ $=$ $A(b -a)$.
Theorem 7 :
If $f$ is of bounded variation on $[a, b]$, say $Σ$$|Δf$$k$$|$$≤$ $M$ for all partitions of $[a, b]$, then $f$ is bounded on $[a, b]$. In fact,
$|f(x)|$ $≤$ $|f(a)|$+ $M$ for all x in $[a, b]$.
$Proof$: Assume that $x ∈ (a, b)$. Using the special partition $P = {a, x, b}$, we find
$|f(x) – f(a)| + |f(b) – f(x)|$ $≤$ $M$.