Functions of Bounded Variation
We know that if f is integrable, then the lower and upper sums of every partition F approximate its integral from below and above, and so the difference between either sum and the integral is at most \(S_{F} - s_{F} =\varOmega _{F}\) , the oscillatory sum
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Functions of Bounded Variation
We know that if f is integrable, then the lower and upper sums of every partition F approximate its integral from below and above, and so the difference between either sum and the integral is at most SF − sF = ΩF , the oscillatory sum corresponding to F. Thus the oscillatory sum is an upper bound for the difference between the approximating sums and the integral. We also know that if f is integrable, then the oscillating sum can become smaller than any fixed positive number for a sufficiently fine partition (see Theorem 14.23). If the function f is monotone, we can say more: ΩF ( f ) ≤ | f (b) − f (a)| · δ (F) for all partitions F, where δ (F) denotes the mesh of the partition F (see Theorem 14.28 and inequality (14.19) in the proof of the theorem). A similar inequality holds for Lipschitz functions: if | f (x) − f (y)| ≤ K · |x − y| for all x, y ∈ [a, b], then ΩF ( f ) ≤ K ·(b−a)· δ (F) for every partition F (seeExercise 17.1). We can state this condition more concisely by saying that ΩF ( f ) = O δ (F) holds for f if there exists a number C such that for an arbitrary partition F, ΩF ( f ) ≤ C · δ (F). (Here we used the big-oh notation seen on p. 141.) By the above, this condition holds for both monotone and Lipschitz functions. Is it true that the condition ΩF ( f ) = O δ (F) holds for every integrable function? The answer is no: one can show that the function x · sin(1/x), if 0 < x ≤ 1; f (x) = 0, if x = 0 does not satisfy the condition (see Exercise 17.3). It is also true that for an arbitrary sequence ωn that tends to zero, there exists a continuous function f : [0, 1] → R such that ΩFn ( f ) ≥ ωn for all n, where Fn denotes a partition of [0, 1] into n equal subintervals (see Exercise 17.4). That is, monotone functions “are better behaved” than continuous functions in this aspect. We characterize below the class of functions for which ΩF ( f ) = O δ (F) holds. By what we stated above, every monotone and every Lipschitz function is included © Springer New York 2015 M. Laczkovich, V.T. S´os, Real Analysis, Undergraduate Texts in Mathematics, DOI 10.1007/978-1-4939-2766-1 17
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17 Functions of Bounded Variation
in this class, but not every continuous function is. The elements of this class are the so-called functions of bounded variation, and they play an important role in analysis. Definition 17.1. Let the function f be defined on the interval [a, b]. If we have a partition of the interval [a, b] given by F : a = x0 < x1 < · · · < xn = b, let VF ( f ) denn
ote the sum ∑ | f (xi ) − f (xi−1 )|. The total variation of f over [a, b] is the supremum i=1
of the set of sums VF ( f ), where F ranges over all partitions of the interval [a, b]. We denote the total variation of f on [a, b] by V ( f ; [a, b]) (which can be infinite). We say that the function f : [a, b] → R is of bounded variation if V ( f ; [a, b]) < ∞. Remarks 17.2. 1. Suppose that the graph of the function f consists of finitely many monotone segments. Let f be monotone on each of the intervals [c
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