Real-Analysis Functions of-Bounded Variation, guaranteed and verified 100% Pass

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Functions of Bounded Variation


By Lebesgue’s theorem we know that a monotonic function on an open interval is
differentiable a.e.. Hence a function that is the difference of two increasing (or
decreasing) functions is also differentiable a.e.. We now want to characterize the
class of functions on a closed, bounded interval which are the difference of two
increasing (or decreasing) functions.



Def. Let 𝑓 be a real valued function defined on a closed, bounded interval [𝑎, 𝑏]
and 𝑃 a partition {𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑘 } of [𝑎, 𝑏]. The variation of 𝒇 with respect
to 𝑷 is defined as:

𝑉(𝑓, 𝑃) = ∑𝑘𝑖=1|𝑓(𝑥𝑖 ) − 𝑓(𝑥𝑖−1 )|.

Length=|𝑓(𝑥𝑖 ) − 𝑓(𝑥𝑖−1 )|

𝑦 = 𝑓(𝑥)




𝑥𝑖−1 𝑥𝑖




The total variation of 𝒇 on [𝑎, 𝑏] is defined as:

𝑇𝑉 (𝑓 ) = sup{𝑉(𝑓, 𝑃)| 𝑃 𝑎 𝑝𝑎𝑟𝑡𝑖𝑡𝑖𝑜𝑛 𝑜𝑓 [𝑎, 𝑏]} .

, 2


Def. A real valued function 𝑓 on the closed, bounded interval [𝑎, 𝑏] is said to be
of bounded variation if 𝑇𝑉 (𝑓 ) < ∞.



Ex. If 𝑓 is an increasing function on [𝑎, 𝑏], then 𝑓 is of bounded variation and
𝑇𝑉 (𝑓) = 𝑓(𝑏) − 𝑓(𝑎).


Given any partition 𝑃 of [𝑎, 𝑏]:

𝑉 (𝑓, 𝑃) = ∑𝑘𝑖=1 |𝑓(𝑥𝑖 ) − 𝑓(𝑥𝑖−1 )|
= ∑𝑘𝑖=1(𝑓 (𝑥𝑖 ) − 𝑓(𝑥𝑖−1 )) = 𝑓(𝑏) − 𝑓(𝑎).

Thus 𝑇𝑉 (𝑓 ) = sup 𝑉 (𝑓, 𝑃) = 𝑓 (𝑏) − 𝑓(𝑎).
𝑃



Ex. Let 𝑓 be a Lipschitz function on [𝑎, 𝑏]. Then 𝑓 is of bounded variation on
[𝑎, 𝑏] and 𝑇𝑉(𝑓) ≤ 𝑐(𝑏 − 𝑎) where c is the Lipschitz constant,
|𝑓(𝑢) − 𝑓(𝑣)| ≤ 𝑐|𝑢 − 𝑣| for all 𝑢, 𝑣 ∈ [𝑎, 𝑏].


Let 𝑃 = {𝑥0 , 𝑥1 , 𝑥2 , … , 𝑥𝑘 } be any partition of [𝑎, 𝑏]. Then:

𝑉(𝑓, 𝑃) = ∑𝑘𝑖=1 |𝑓 (𝑥𝑖 ) − 𝑓 (𝑥𝑖−1 )| ≤ ∑𝑘𝑖=1 𝑐 |𝑥𝑖 − 𝑥𝑖−1 | = 𝑐|𝑏 − 𝑎|.


Thus 𝑐|𝑏 − 𝑎| is an upper bound for 𝑉(𝑓, 𝑃) and 𝑇𝑉(𝑓) ≤ 𝑐(𝑏 − 𝑎).