Absolute Continuity But Not Bounded Variation

Absolute Continuity but Not Bounded Variation: A Deep Dive

This article explores the fascinating relationship between absolute continuity and bounded variation in the context of real analysis. While seemingly similar, these concepts differ significantly, and understanding their nuances is crucial for a solid grasp of measure theory and integration. We'll delve into the definitions, provide illustrative examples, and clarify why a function can be absolutely continuous without possessing bounded variation.

What is Absolute Continuity?

A function f defined on a closed interval [a, b] is said to be absolutely continuous if for every ε > 0, there exists a δ > 0 such that for any finite collection of disjoint subintervals [x_i, y_i] of [a, b], the following condition holds:

∑_i |y_i - x_i| < δ implies ∑_i |f(y_i) - f(x_i)| < ε

This definition essentially states that the total variation of f over a collection of small intervals can be made arbitrarily small by choosing sufficiently small intervals. This is a stronger condition than mere continuity; absolute continuity implies continuity, but the converse is not true. Absolute continuity is deeply connected to the concept of the Lebesgue integral. A function is absolutely continuous if and only if it is the indefinite integral of an integrable function.

What is Bounded Variation?

A function f defined on a closed interval [a, b] is said to have bounded variation if its total variation is finite. The total variation, denoted by V_a^b(f), is defined as the supremum of the sums:

V_a^b(f) = sup {∑_i=1ⁿ |f(x_i) - f(x_i-1)| : a = x₀ < x₁ < ... < x_n = b}

Essentially, a function has bounded variation if the total change in its value over all possible partitions of the interval [a, b] is finite. This reflects the "oscillation" of the function. A function with bounded variation can be decomposed into the difference of two monotonically increasing functions.

The Key Difference: Why Absolute Continuity Doesn't Imply Bounded Variation

While absolute continuity implies continuity, it does not imply bounded variation. The crucial distinction lies in the way the variation is controlled. Absolute continuity controls the variation over collections of arbitrarily small intervals, while bounded variation controls the variation over all possible partitions of the interval. A function can have large oscillations concentrated in a small region, rendering its total variation unbounded, even if its variation over small collections of intervals remains under control.

Example: A Function That's Absolutely Continuous But Not of Bounded Variation

Consider the function defined on the interval [0, 1]:

f(x) = x sin(1/x) for x ≠ 0, and f(0) = 0.

This function is absolutely continuous. However, it's not of bounded variation. The oscillations near x = 0 become increasingly frequent and intense, leading to an unbounded total variation. Although the variation over any small collection of intervals can be controlled, the cumulative variation over all partitions diverges. This showcases the stronger nature of the absolute continuity criterion compared to bounded variation.

Conclusion:

Absolute continuity and bounded variation are distinct but related concepts in real analysis. Absolute continuity is a stronger condition, implying continuity but not necessarily bounded variation. Understanding this distinction is vital for a deeper comprehension of measure theory, integration, and the behavior of functions in advanced calculus. The example provided highlights the crucial difference and illustrates how a function can satisfy the constraints of absolute continuity while exhibiting unbounded total variation.