18.099b Problem Set 7a (Corrected) Due: Thursday, April 29th.

This assignment is to write an essay on functions of bounded variation. It should be typeset in Tex. Arrange and express the material discussed below in whatever way you think best. The paper should flow smoothly and be well motivated.

Suppose a ≤ b are real numbers. Let P[a, b] denote the set of all partitions of the interval [a, b]. Suppose f(x) is a function defined on an interval [a, b]. For any P ∈ P[a, b] denote by S_{f}(P) the sum

S_{f}(P) := ∑_{k=1}^{n} |f(x_{k}) − f(x_{k−1})|

where P is the partition a = x_{0} < x_{1} < x_{2} < ⋯ < x_{n−1} < x_{n} = b. The total variation of f(x) on [a, b] is defined to be

V_{f}(a, b) ≡ sup{S_{f}(P) : P ∈ P[a, b]}.

That is, V_{f}(a, b) is the supremum of the set of all S_{f}(P) for all possible partitions P of [a, b]. If V_{f}(a, b) < ∞, then f is said to be of bounded variation on [a, b].

Show that all monotonic functions are of bounded variation. In fact, the total variation of a monotonic function on [a, b] is |f(b) − f(a)|.

Note that functions of bounded variation are bounded. However, there are functions that are bounded, and even continuous on a closed interval, but are not of bounded variation. Hint: Consider x sin(\frac{1}{x}) on [0, 1].

Let BV[a, b] denote the set of functions of bounded variation on [a, b]. Show that BV[a, b] is a linear space: If f, g ∈ BV[a, b] and c ∈ ℝ, then f + g ∈ BV[a, b] and cf ∈ BV[a, b].

A function f on a closed interval [a, b] is said to satisfy a Lipschitz condition if there exists a constant K such that

|f(x) − f(y)| ≤ K|x − y|

for all x, y ∈ [a, b]. Show that if f satisfies a Lipschitz condition then it is of bounded variation. Use this to conclude that if f(x) has a derivative at every point on [a, b] and if f′(x) is bounded on [a, b], then f ∈ BV[a, b]. Hint: Mean value theorem.

As an aside, discuss contractions, which are a special case of functions satisfying a Lipschitz condition. A function f on a closed interval [a, b] is a contraction if there exists a constant λ < 1 such that

|f(x) − f(y)| ≤ λ|x − y|

for all x, y ∈ [a, b]. Prove the fixed-point theorem which states that if f is a contraction on [a, b] with values in [a, b], then it has a unique fixed-point in [a, b]; that is, there exists a unique number x ∈ [a, b] such that f(x) = x.

Getting back to bounded variation, prove the following striking fact: A function f(x) defined on [a, b] is of bounded variation if and only if it is the difference of two increasing functions on [a, b]. Hint: Suppose f ∈ BV[a, b]. Show that for any x ≤ x′ from [a, b],

V_{f}(a, x′) ≥ V_{f}(a, x) + |f(x) − f(x′)|.

Conclude that the function g(x) = V_{f}(a, x) and the function h(x) = f(x) + V_{f}(a, x) are both increasing on [a, b].

Is a representation of f ∈ BV[a, b] as a difference of two increasing functions unique? Give several examples.

Prove that functions of bounded variation are integrable.