Functions of Bounded Variation — Mathematics Knowledge System

Overview

A function f on [a,b] has bounded variation if the total up-and-down 'wiggle' of its graph — measured by summing |f(xᵢ₊₁)-f(xᵢ)| over a partition and taking the supremum over all partitions — is finite. Introduced by Camille Jordan in 1881 while studying the convergence of Fourier series, the class of functions of bounded variation (BV) sits strictly between continuous functions and arbitrary bounded functions: every monotone function and every function with a continuous derivative on a compact interval is BV, but BV functions can still have jump discontinuities (just only countably many, and each of finite size). The single most important structural fact is the Jordan decomposition: every BV function can be written as the difference of two monotone increasing functions — reducing many questions about general BV functions to the much simpler monotone case.

Intuition

Imagine tracing the graph of f with a pen, moving only left to right, but allowed to move up and down as much as needed. The total variation is the total vertical distance your pen travels — add up every uphill climb and every downhill drop, ignoring horizontal motion entirely. A function is 'bounded variation' if this total ink-vertical-travel is finite, no matter how many wiggles the function has. A monotone function (always climbing or always falling) has the smallest possible total variation for its endpoints — just |f(b)-f(a)| — while a wildly oscillating function forces the pen to retrace much more vertical distance than the net change from f(a) to f(b), inflating the total variation far above |f(b)-f(a)|.

Formal Definition

Definition

For a partition P: a=x₀<x₁<⋯<xₙ=b of [a,b], define the variation of f over P, then the total variation as the supremum over all partitions:

V(f, P) = \sum_{i=0}^{n-1} |f(x_{i+1}) - f(x_i)|

Variation of f over the partition P

V_a^b(f) = \sup_{P} V(f,P)

The supremum is taken over ALL possible partitions of [a,b]

Total variation of f on [a,b]

f \in BV[a,b] \iff V_a^b(f) < \infty

Definition: f has bounded variation

Notation

Notation	Meaning
$V(f,P) = \sum_i \|f(x_{i+1})-f(x_i)\|$	The variation of f over a specific partition P — the sum of absolute increments
$V_a^b(f)$	The total variation of f on [a,b]; the supremum of V(f,P) over all partitions P
$BV[a,b]$	The set (vector space) of functions of bounded variation on [a,b]

Derivation

Proof sketch of the Jordan decomposition: writing f as the difference of the running total variation and a complementary monotone function.

\text{Define } g(x) = V_a^x(f) \text{ (the total variation of } f \text{ restricted to } [a,x]).

The running total variation, as x ranges from a to b

\text{For } x_1 < x_2: \quad g(x_2) - g(x_1) = V_{x_1}^{x_2}(f) \ge |f(x_2)-f(x_1)| \ge f(x_2)-f(x_1)

By additivity of variation over sub-intervals, and the reverse triangle inequality

\implies g(x_2) - f(x_2) \ge g(x_1) - f(x_1)

Rearranging shows h := g - f is monotone increasing

g \text{ is monotone increasing (trivially, since variation only accumulates)}, \quad h = g - f \text{ is monotone increasing (just shown)}

Both pieces are monotone

f = g - h

Recovering f as the difference of two monotone increasing functions completes the decomposition

Properties

Monotone functions are BV

\text{If } f \text{ is monotone on } [a,b], \text{ then } f \in BV[a,b] \text{ and } V_a^b(f) = |f(b)-f(a)|.

Condition: For a monotone function, every increment f(xᵢ₊₁)-f(xᵢ) has the same sign, so the sum telescopes exactly to |f(b)-f(a)| regardless of the partition.

C¹ functions are BV

\text{If } f \text{ has a continuous derivative on } [a,b], \text{ then } f \in BV[a,b] \text{ and } V_a^b(f) = \int_a^b |f'(x)|\,dx.

Jordan decomposition

f \in BV[a,b] \iff f = g - h \text{ for some monotone increasing } g, h \text{ on } [a,b].

Condition: Take g(x) = V_a^x(f) (the running total variation) and h = g - f; both are monotone increasing, and their difference recovers f exactly.

BV functions have only countably many discontinuities

\text{If } f \in BV[a,b], \text{ its set of discontinuities is countable, and each is a jump discontinuity (one-sided limits exist).}

Condition: Follows from the Jordan decomposition: monotone functions can only have jump discontinuities, and only countably many of them.

Additivity over sub-intervals

V_a^b(f) = V_a^c(f) + V_c^b(f) \quad (a<c<b)

BV implies bounded

f \in BV[a,b] \implies f \text{ is bounded on } [a,b].

Condition: |f(x)| \le |f(a)| + V_a^b(f) for all x, since the variation controls how far f can stray from f(a).

Applications

Convergence of Fourier series at a point depends on the local behavior of the function; Jordan's classical theorem states that if f is of bounded variation near a point x, its Fourier series at x converges to the average of the left and right limits — the historical motivation for defining BV functions in the first place.

Worked Examples

f is monotone increasing (slope 3>0), so the total variation equals |f(2)-f(0)|.
$f(0) = 1, \quad f(2) = 7$
Apply the monotone-function formula.
$V_0^2(f) = |f(2)-f(0)| = |7-1| = 6$

Answer: 6

Practice Problems

Difficulty 5/10

Find the total variation of f(x) = 5 - 2x on [1,4] (a monotone decreasing linear function).

Difficulty 6/10

Which of the following is guaranteed by the Jordan decomposition theorem?

Every BV function is continuousEvery BV function is the difference of two monotone increasing functionsEvery BV function is differentiableEvery continuous function is BV

Difficulty 8/10

Prove that every function f with a continuous derivative on [a,b] is of bounded variation, with V_a^b(f) = ∫ₐᵇ|f'(x)|dx.

Common Mistakes

Common Mistake

Believing continuity implies bounded variation.

False — f(x) = x sin(1/x) (extended by f(0)=0) is continuous on [0,1] but has infinite total variation, because it oscillates infinitely often near 0 with amplitude shrinking too slowly (only linearly) to keep the total variation finite.

Common Mistake

Computing total variation as simply |f(b) - f(a)|, regardless of the function's shape.

|f(b)-f(a)| is only the total variation when f is monotone. If f goes up and then down (or vice versa), the total variation must sum the absolute change over EACH monotone piece — as in f(x)=x²-2x on [0,3], where the correct total variation (5) is larger than |f(3)-f(0)| (which is only 3).

Common Mistake

Assuming BV functions cannot have discontinuities.

BV functions can have jump discontinuities — the Jordan decomposition (as a difference of monotone functions) guarantees only that these discontinuities are countable in number and are simple jumps (one-sided limits exist), not that there are none.

Quiz

Flashcards

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Summary

f has bounded variation on [a,b] if V_a^b(f) = sup over partitions of Σ|f(xᵢ₊₁)-f(xᵢ)| is finite.
Monotone functions are BV with V_a^b(f) = |f(b)-f(a)|; C¹ functions are BV with V_a^b(f) = ∫ₐᵇ|f'(x)|dx.
The Jordan decomposition writes any BV function as a difference of two monotone increasing functions, reducing general BV analysis to the monotone case.
BV functions can have jump discontinuities, but only countably many, each a simple jump (one-sided limits exist) — a consequence of the Jordan decomposition.
Continuity does not imply bounded variation: f(x)=x sin(1/x) (f(0)=0) is continuous on [0,1] but has infinite total variation, since its oscillations near 0 don't shrink fast enough.

References

WebsiteWikipedia — Bounded variation
BookRudin, W. Principles of Mathematical Analysis, 3rd ed. Ch. 6.