Question

The oscillation of a bounded function f on an interval [a,b]is the ______of the set {|f(x)-f(y)|}of the numbers.​

Answer 1

. Let f : [a, b] → R be a function and let c ∈ (a, b). If f is of

bounded variation on [a, c] and [c, b], then f is of bounded variation on [a, b] and

V (f, [a, b]) = V (f, [a, c]) + V (f, [c, b])

A proof of Theorem 2.5 is provided in Gordon’s text [1].

2.4. Examples. We now consider some functions that are not of bounded variation on a particular interval. These examples are helpful for gaining insight into

what kinds of functions we might expect not to be of bounded variation, and for

examining the mechanics of showing that a particular function is not of bounded

variation.