Functions of Bounded Variation Example 42

Functions of Bounded Variation

Example 42
Consider the following functions on the given intervals:
(1)
(2)

(3)

Crudely speaking, a function is said to be of bounded variation if the amount of oscillation (i.e.,

wiggle) in its graph is manageable (i.e., tame) in some sense over a particular interval. We will see

later that the functions (1) and (2) in Example 42 are functions of bounded variation on while

the function in (3) simply wiggles too wildly on . We are interested in the notion of bounded

variation as it provided us with yet another descriptor for a function (e.g., continuous, differentiable,
monotonic, even/odd, Lipschitz, .. . . ). Functions of bounded variation also play a central role in

the study of integration theory.

Definition 58 is a finite collection with
(1) A partition - of for

. .

(2) We say that a partition - of is regular if

. a refinement of the partition - if
(3) We call the partition Q of

A partition may be viewed either as a finite collection of points or as a collection of subintervals
. We note that when regarding a partition as a collection of

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subintervals that two adjacent subintervals have a common endpoint. The length of a subinterval
is ; We call the length of the longest subinterval the mesh of - write

. (We note that the mesh of - is a well-defined notion.)

If P and Q are two partitions of [a,b], we call T a common refinement of P and Q provided
that . (Is there such a thing as a least common refinement?)

Example 43
Let . Then

(1) P = {1, 3.5, 4, 6} and R = {1, 2.22, 3.6, 5.99, 5.995, 6} are partitions of I with

and .

(2) Q = {1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6} is a regular partition of I.

(3) Q is a refinement of P as . (In general, what can be said about and

when Q is a refinement of P?)

Definition 59 is said to be increasing on if for all with
A function
we have that . Further, f is said to be strictly increasing if
.

Of course, there is a dual to Definition 59 for decreasing and strictly decreasing functions. (We

omit this statement as it is surely familiar to the reader.) A function that is either increasing or

decreasing on is said to be monotone there.

Theorem 60 and differentiable on .
Suppose that f is continuous on .
.
(1) If on , then f is strictly increasing on

(2) If on , then f is a constant function on

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(3) If on , then f is strictly decreasing on .

Proof . Applying the Mean Value Theorem (Corollary 53) to f on
(1)/(3) Suppose

we obtain

for some . Since with , (1) - (3) follow from (*) >

Example 44 is required for condition (2) in the above theorem. Define
Show that the continuity of f on
f on as follows:

The above function is differentiable on with there and, hence, is continuous there

yet f is not identically equal to a constant there. In fact, f is increasing on .

Example 45 - Cantor ternary function

Let . Write x as where ; That is, write x as a base 3 decimal.

If possible, use no 1's in the ternary expansion of x. Here are a few conversions: ,
,, , ,

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and instead

of . (Note: The Mathematica command BaseForm[x,3] converts the

base 10 number x to base 3.) For a given , let equal

4 if no and otherwise let be the smallest value such that . Using the above

conversions we have , , , , ,

and . Finally, define the Cantor ternary function as follows:

.

The above is not nearly as complex as it might initially seem. For example, we have that

,, ,, ,

and . The graph of follows:

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Properties of the Cantor ternary function: to itself with &.
(1) is a surjective mapping of

(2) is monotone increasing on .

(3) is (uniformly) continuous.

(4) The arclength of the curve from to is 2. (Hence, the usual

arclength formula from calculus fails as “almost everywhere”.)

We leave the verification of the above to the interested student.

Recall: and

Example 46

Suppose that . Find for .

Observe that the difference yields the magnitude of the jump discontinuities at 1 &

5. Further observe that this difference is zero at points where f is continuous. In general, if
, we say that f has a jump discontinuity at .

Theorem 61 and let be any partition
Suppose that f is increasing on

of . Then

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.

Proof and . For let be any point in
Set

. Since f is increasing on , we have that for

. Hence, for we have that

.

Now,

A similar result holds for decreasing functions.

Theorem 62 , then the set of discontinuities is at most countable.
If the function f is monotone on

Proof . For each , let .
Assume f is increasing on

If , then by Theorem 61 we have that

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or elements. Hence, the set
.

We conclude that each set is a finite set with at most

is at most countable being the countable union of finite sets. Since the only discontinuities

for the monotone function f are jump discontinuities, the set is equal to the set of

discontinuities for f. The decreasing case can be established by considering the function
which is increasing when f is decreasing. >

Definition 63 (Camille Jordan 1838-1922)

Suppose . If is any partition of , define

.

We define the total variation of f on by

where the supremum is taken over all possible partitions of . In the case , we

say that f is of bounded variation on .

The collection of all functions of bounded variation on is denoted by . It is left as

an exercise to show that the collection is a vector space over ú. It is also an easy exercise

to show that functions of bounded variation on are, in fact, also bounded there.

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Theorem 64 , then f is of bounded variation there with .
If f is monotone of

Proof
Clear.

Of course, in general, . This can be establish by considering the trivial

partition of .

Theorem 65 are continuous on ), then f is of bounded variation there.
If (that is,

Proof , so is the function . Applying the Extreme Value Theorem
Since is continuous on

to there exists M ( = absolute maximum of on ) so that . Let

be any partition of . Then by the Mean-Value Theorem

for each there exists such that

.

It follows that

and so

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Hence, and the function f is of bounded variation on .>

We note that is of bounded variation on since it strictly increasing there yet

its derivative is unbounded on .

Example 47 on . Then
Let

with for . It follows from Theorem 65 that f is of bounded
variation on with . Consider the graph of f below:

Clearly, while f has some “wiggle” on , it does not wiggle too much (although the range of

f is rather large!). We note that f has local extrema at 2, 4 and 8. Can we do better than

? Let be a regular

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partition on . Then, using Mathematica, , ,
,, , and

. The total variation of f is .

It can be shown that if , then . In general, it is the case that

. (The Cantor ternary function is a function that yields strict inequality
although this is beyond the scope of this class.)

Example 48

Let . If we define , then and

so .
where is as defined above, then is a partition of
If
and we have that

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As , and so the function f is not of bounded variation on .
Example 49
Let . Then which

is continuous on . So, by Theorem 65, f is of bounded variation on with
.

Definition 66 if there exists such
We say a function f satisfies a Lipschitz condition on
that

for all .

The above notion is named for the German mathematician Rudolf Lipschitz (1832-1903). By
rewriting the condition

as

we see that a function is Lipschitz on provided that the slopes of all secant lines are bounded
by . That is, for a fixed value the graph of f lies between the two lines

and
.

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This is sometimes referred to as a cone condition.

Illustration on . Then one can show that
Suppose that

for all . The graphic below shows the “cones” at :

The slopes of the sides on the above cones are .

Proposition 67
If , then f satisfies a Lipschitz condition there.

The proof of the above follows easily from the Mean-Value Theorem.

Example 50

(1) satisfies a Lipschitz condition with on any non-degenerate closed

interval (Proposition 67) yet it is not of bounded variation on ú.

(2) satisfies a Lipschitz condition on with .

(Proposition 67)

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(3) fails to satisfy a Lipschitz condition on ú.

(4) satisfies a Lipschitz condition on with

. (Can we do “better” than ? Think cone condition.) Observe that this

function is not in yet is Lipschitz. The collection of functions satisfying a

Lipschitz condition on is a superset of .

(5) fails to satisfy a Lipschitz condition on any closed, bounded interval
containing 2. Why? (Think cone condition.)

Theorem 68 , then f is of bounded variation there.
If f satisfies a Lipschitz condition on

Proof be any partition of . Then
Let

Hence, by the Completeness Axiom, and the function f is of bounded
and
variation on .>

Theorem 69 , then ,
If and .

Proof be any partition of and let
Let

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be any partition of . Then is a

partition of with

.

Fix the partition and let vary over all possible partitions of . It follows that

and so f is of bounded variation on .

Letting vary over all possible partitions of we have that and

.

We now establish the reverse of the above inequality. Let be any

partition of . WLOG, we may assume that with, say, . Then

is a partition of and

is a partition of . We now have that

.

Letting - vary over all partitions of we see that
.

We conclude that .>

Theorem 70 and we define . Then the function is increasing on
Suppose that

.

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Proof is well-defined by virtue of Theorem 69. Suppose that with
The function .

. By Theorem 69,

and so . .>
Thus, and is increasing on

Recall that if , then . So, in this case,

Example 51 . Then one can show that
(1) Suppose .
are shown below:
The graph of

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(2) Let on . Then
. The graph of
and where for

are shown below:

(3) Suppose that on . Here . The graph of
are shown below:

Here we have that .

Proposition 71 and we define . Then f is continuous at x iff is
Suppose that

continuous at x.

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See page 530 in Lebesgue Integration on Euclidean Space by Frank Jones for a proof of the above.

Theorem 72 and we define . Then the function is increasing
Suppose that

on .

Proof . Consider
Let

In general, it is true that (See the comment after Theorem 64.) Hence,

and so . .>
Thus, the function is increasing on

In some sense we now state and prove the capstone result of this unit.

Theorem 73 - Jordan Decomposition Theorem

Suppose that . Then f is bounded variation on iff f can be written as the

difference of two monotone increasing functions.

Proof are two monotone increasing functions with . By
Suppose that
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Theorem 64, and since is a vector space over ú, it follows that
Suppose
.

. By Theorem 70, is increasing on . By Theorem 72,

is increasing on . Since , f can be written as the difference of two

increasing functions. (We note that the representation of a function of bounded variation as the
difference of two increasing functions is not unique.) >

The last result is an immediate consequence of Theorems 62 & 73.

Proposition 74 .
If , then f is continuous except on at most a countable subset of

A function of bounded variation has only simple discontinuities (either removable or jump
discontinuities).

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