Question

Prove every absolute continuous function is bounded variation

Answer 1

 is said to be absolutely continuous on [a,b] if for all ϵ>0there exists a δ>0 such that for all finite collections of disjoint open subintervals of [a,b], {(x1,y1),(x2,y2),...,(xn,yn)} with ∑i=1n(yi−xi)we have that:

(1)

∑i=1n|f(yi)−f(xi)|<ϵ

We noted that every absolutely continuous function f on [a,b]is uniformly continuous on [a,b] and hence continuous on [a,b].

We now show that every absolutely continuous function fon [a,b] is of bounded variation on [a,b].

Theorem 1: If f:[a,b]→R is absolutely continuous on [a,b] then f is of bounded variation on [a,b].Since f is absolutely continuous on [a,b] for ϵ=1>0 there exists a δ>0 such that for every finite collection of disjoint open subintervals of [a,b], {(x1,y1),(x2,y2),...,(xn,yn)} with ∑i=1n(yi−xi)<δ we have that:(2)

∑i=1n|f(yi)−f(xi)|<ϵ=1

Let P∗={a=a0,a1,...,an=b} be a partition of [a,b] with the property that ai−ai−1=δ2 for all i∈{1,2,...,n−1}, and such that an−an−1≤δ2.



Then we have that:(3)

n=⌊2(b−a)δ⌋+1

(This is because the length of the interval (a,b) is b−a. We then divide this length by δ2 and take the floor of this number. This gives us the number of times δ2 goes into b−a. We add +1 to account for the remaining interval (an−1,an)whose less is less than or equal to δ2.)Now let P be any partition of [a,b] and let:(4)

P′=P∪P∗

Then P′ is a refinement of P and P∗. Let P′={z0=a,z1,...,zm=n}. For each i∈{1,2,...,n} let:(5)

P′i={zik∈P′:zik∈[ai−1,ai]}

That is, P′i is the sets of points in Pi that are contained in the closed interval [ai−1,ai] from the partition P∗. Then:(6)

V(P,f)≤V(P′,f)=∑i=1n∑k|f(zik)−f(zik−1)|≤∑k=1n1=n=⌊2(b−a)δ⌋+1

Therefore the variation of V(P,f) is always bounded by ⌊2(b−a)δ⌋+1, so f is of bounded variation on [a,b].

Answer 2

We noted that every absolutely continuous function f on [a,b] is uniformly continuous on [a,b] and hence continuous on [a,b].

We now show that every absolutely continuous function f on [a,b] is of bounded variation on [a,b]