Seja
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
f
{\displaystyle f\,}
[
a
,
b
]
{\displaystyle [a,b]\,}
D
{\displaystyle D\,}
osc
[
a
,
b
]
(
f
)
:=
sup
[
a
,
b
]
f
(
x
)
−
inf
[
a
,
b
]
f
(
x
)
{\displaystyle {\hbox{osc}}_{[a,b]}(f):=\sup _{[a,b]}f(x)-\inf _{[a,b]}f(x)\,}
Se
f
{\displaystyle f\,}
osc
[
a
,
b
]
(
f
)
=
f
(
b
)
−
f
(
a
)
{\displaystyle {\hbox{osc}}_{[a,b]}(f)=f(b)-f(a)\,}
Se
f
{\displaystyle f\,}
osc
[
a
,
b
]
(
f
)
=
f
(
a
)
−
f
(
b
)
{\displaystyle {\hbox{osc}}_{[a,b]}(f)=f(a)-f(b)\,}
editar
Seja
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
f
{\displaystyle f\,}
P
:=
{
x
0
:=
a
,
x
1
,
x
2
,
…
,
x
n
−
1
,
x
n
:=
b
}
{\displaystyle P:=\left\{x_{0}:=a,x_{1},x_{2},\ldots ,x_{n-1},x_{n}:=b\right\}\,}
[
a
,
b
]
{\displaystyle [a,b]\,}
D
{\displaystyle D\,}
var
P
(
f
)
:=
∑
i
=
1
n
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
{\displaystyle {\hbox{var}}_{P}(f):=\sum _{i=1}^{n}\left|f(x_{i})-f(x_{i-1})\right|\,}
1. Seja P uma partição cujos extremos são
x
0
=
a
{\displaystyle x_{0}=a\,}
x
n
=
b
{\displaystyle x_{n}=b\,}
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
D
⊃
[
a
,
b
]
{\displaystyle D\supset [a,b]\,}
var
P
(
α
f
)
=
|
α
|
var
P
(
f
)
,
∀
α
∈
R
{\displaystyle {\hbox{var}}_{P}(\alpha f)=|\alpha |{\hbox{var}}_{P}(f),~~\forall \alpha \in \mathbb {R} \,}
Demonstração Imediato da definição.
2. Seja P uma partição cujos extremos são
x
0
=
a
{\displaystyle x_{0}=a\,}
x
n
=
b
{\displaystyle x_{n}=b\,}
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
D
⊃
[
a
,
b
]
{\displaystyle D\supset [a,b]\,}
var
P
(
f
)
=
|
f
(
a
)
−
f
(
b
)
|
{\displaystyle {\hbox{var}}_{P}(f)=\left|f(a)-f(b)\right|\,}
Demonstração
Considere, sem perda de generalidade, que
f
{\displaystyle f\,}
var
P
(
f
)
:=
∑
i
=
1
n
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
{\displaystyle {\hbox{var}}_{P}(f):=\sum _{i=1}^{n}\left|f(x_{i})-f(x_{i-1})\right|\,}
Como
x
i
≥
x
i
−
1
{\displaystyle x_{i}\geq x_{i-1}\,}
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
=
f
(
x
i
)
−
f
(
x
i
−
1
)
≥
0
{\displaystyle \left|f(x_{i})-f(x_{i-1})\right|=f(x_{i})-f(x_{i-1})\geq 0\,}
var
P
(
f
)
:=
∑
i
=
1
n
f
(
x
i
)
−
f
(
x
i
−
1
)
=
f
(
x
n
)
−
f
(
x
0
)
=
f
(
b
)
−
f
(
a
)
=
|
f
(
a
)
−
f
(
b
)
|
{\displaystyle {\hbox{var}}_{P}(f):=\sum _{i=1}^{n}f(x_{i})-f(x_{i-1})=f(x_{n})-f(x_{0})=f(b)-f(a)=\left|f(a)-f(b)\right|\,}
3. Seja P uma partição cujos extremos são
x
0
=
a
{\displaystyle x_{0}=a\,}
x
n
=
b
{\displaystyle x_{n}=b\,}
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
g
:
D
→
R
{\displaystyle g:D\to \mathbb {R} \,}
D
⊃
[
a
,
b
]
{\displaystyle D\supset [a,b]\,}
var
P
(
f
+
g
)
≤
var
P
(
f
)
+
var
P
(
g
)
{\displaystyle {\hbox{var}}_{P}(f+g)\leq {\hbox{var}}_{P}(f)+{\hbox{var}}_{P}(g)\,}
Demonstração
var
P
(
f
+
g
)
:=
∑
i
=
1
n
|
f
(
x
i
)
+
g
(
x
1
)
−
f
(
x
i
−
1
)
−
g
(
x
i
−
1
)
|
≤
∑
i
=
1
n
(
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
+
|
g
(
x
i
)
−
g
(
x
i
−
1
)
|
)
=
var
P
(
f
)
+
var
P
(
g
)
{\displaystyle {\begin{array}{rcl}{\hbox{var}}_{P}(f+g)&:=&\sum _{i=1}^{n}\left|f(x_{i})+g(x_{1})-f(x_{i-1})-g(x_{i-1)}\right|\\&\leq &\sum _{i=1}^{n}\left(\left|f(x_{i})-f(x_{i-1})\right|+\left|g(x_{i})-g(x_{i-1})\right|\right)\\&=&{\hbox{var}}_{P}(f)+{\hbox{var}}_{P}(g)\end{array}}\,}
4. Seja P uma partição cujos extremos são
x
0
=
a
{\displaystyle x_{0}=a\,}
x
n
=
b
{\displaystyle x_{n}=b\,}
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
D
⊃
[
a
,
b
]
{\displaystyle D\supset [a,b]\,}
P' é um refinamento de P
var
P
f
≤
var
P
′
(
f
)
{\displaystyle {\hbox{var}}_{P}f\leq {\hbox{var}}_{P'}(f)\,}
Demonstração
Sem perda de generalidade, considere que P' é um refinamento de P pela inclusão de um único ponto
x
k
−
1
≤
x
′
≤
x
k
{\displaystyle x_{k-1}\leq x'\leq x_{k}\,}
|
f
(
x
k
−
1
)
−
f
(
x
k
)
|
≤
|
f
(
x
k
−
1
)
−
f
(
x
′
)
|
+
|
f
(
x
′
)
−
f
(
x
k
)
|
{\displaystyle \left|f(x_{k-1})-f(x_{k})\right|\leq \left|f(x_{k-1})-f(x')\right|+\left|f(x')-f(x_{k})\right|\,}
o resultado segue.
Seja
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
f
{\displaystyle f\,}
[
a
,
b
]
{\displaystyle [a,b]\,}
D
{\displaystyle D\,}
var
[
a
,
b
]
(
f
)
:=
sup
P
∈
P
var
P
(
f
)
{\displaystyle {\hbox{var}}_{[a,b]}(f):=\sup _{P\in \mathbb {P} }{\hbox{var}}_{P}(f)\,}
O supremo é tomado em
P
{\displaystyle \mathbb {P} \,}
[
a
,
b
]
{\displaystyle [a,b]\,}
As seguintes propriedades são de demonstração imediata, aparir da definição de supremo e das propriedades já demonstradas para a variação em uma partição.
1. Se
f
{\displaystyle f\,}
var
[
a
,
b
]
(
f
)
=
|
f
(
a
)
−
f
(
b
)
|
{\displaystyle {\hbox{var}}_{[a,b]}(f)=\left|f(a)-f(b)\right|\,}
2. Se
f
{\displaystyle f\,}
var
[
a
,
b
]
(
f
)
≥
var
[
c
,
d
]
(
f
)
{\displaystyle {\hbox{var}}_{[a,b]}(f)\geq {\hbox{var}}_{[c,d]}(f)\,}
a
≤
c
≤
d
≤
b
{\displaystyle a\leq c\leq d\leq b\,}
3. Se
f
{\displaystyle f\,}
g
{\displaystyle g\,}
var
[
a
,
b
]
(
f
+
g
)
≤
var
[
a
,
b
]
(
f
)
+
var
[
a
,
b
]
(
g
)
{\displaystyle {\hbox{var}}_{[a,b]}(f+g)\leq {\hbox{var}}_{[a,b]}(f)+{\hbox{var}}_{[a,b]}(g)\,}
4. Se
f
{\displaystyle f\,}
var
[
a
,
b
]
(
α
f
)
=
|
α
|
var
[
a
,
b
]
(
f
)
,
∀
α
∈
R
{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)=|\alpha |{\hbox{var}}_{[a,b]}(f),~~\forall ~\alpha \in \mathbb {R} \,}
5. Se
f
{\displaystyle f\,}
var
[
a
,
c
]
(
f
)
=
var
[
c
,
b
]
(
f
)
+
var
[
b
,
c
]
(
f
)
,
∀
c
∈
[
a
,
b
]
{\displaystyle {\hbox{var}}_{[a,c]}(f)={\hbox{var}}_{[c,b]}(f)+{\hbox{var}}_{[b,c]}(f),~~\forall ~c\in [a,b]\,}
editar
Diz-se que uma função real
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
variação limitada em um intervalo
[
a
,
b
]
{\displaystyle [a,b]\,}
var
[
a
,
b
]
(
α
f
)
≤
∞
{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)\leq \infty \,}
Seja
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
C
1
[
a
,
b
]
{\displaystyle C^{1}[a,b]\,}
var
[
a
,
b
]
(
f
)
=
∫
a
b
|
f
′
(
x
)
|
d
x
{\displaystyle {\hbox{var}}_{[a,b]}(f)=\int _{a}^{b}|f'(x)|dx\,}
Demontração
Primeiro observamos que se
P
{\displaystyle P\,}
[
a
,
b
]
{\displaystyle [a,b]\,}
var
P
f
=
∑
i
=
1
n
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
=
∑
i
=
1
n
|
f
′
(
x
i
∗
)
|
(
x
i
−
x
i
−
1
)
,
x
i
∗
∈
(
x
i
−
1
,
x
i
)
{\displaystyle {\hbox{var}}_{P}{f}=\sum _{i=1}^{n}\left|f(x_{i})-f(x_{i-1})\right|=\sum _{i=1}^{n}\left|f'(x_{i}^{*})\right|(x_{i}-x_{i-1}),~~~x_{i}^{*}\in (x_{i-1},x_{i})\,}
Da definição de variação total, podemos inferir a existência de uma seqüência de partições
P
k
=
(
x
0
k
,
x
1
k
,
…
,
x
n
k
k
)
,
{\displaystyle P_{k}=\left(x_{0}^{k},x_{1}^{k},\ldots ,x_{n_{k}}^{k}\right),\,}
0
≤
var
[
a
,
b
]
(
f
)
−
var
P
k
(
f
)
≤
1
/
k
{\displaystyle 0\leq {\hbox{var}}_{[a,b]}(f)-{\hbox{var}}_{P_{k}}(f)\leq 1/k\,}
Como a variação não decresce com o refino da partição, pode supor que comprimento das partições
P
k
{\displaystyle P_{k}\,}
var
[
a
,
b
]
(
f
)
=
lim
k
→
∞
var
P
k
(
f
)
=
lim
k
→
∞
∑
i
=
1
n
k
|
f
′
(
x
i
k
∗
)
|
(
x
i
k
−
x
i
−
1
k
)
=
∫
a
b
|
f
′
(
x
)
|
d
x
{\displaystyle {\hbox{var}}_{[a,b]}(f)=\lim _{k\to \infty }{\hbox{var}}_{P_{k}}(f)=\lim _{k\to \infty }\sum _{i=1}^{n_{k}}\left|f'(x_{i}^{k}*)\right|(x_{i}^{k}-x_{i-1}^{k})=\int _{a}^{b}|f'(x)|dx\,}
Uma função é de variação limitada se e somente se pode ser escrita como a diferença de duas funções não decrescentes.
Demontração a. Seja
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
D
{\displaystyle D\,}
F
:
D
2
→
R
{\displaystyle F:D^{2}\to \mathbb {R} \,}
F
(
x
0
,
x
)
=
{
var
[
x
0
,
x
]
(
f
)
,
x
0
≤
x
−
var
[
x
0
,
x
]
(
f
)
,
x
0
>
x
{\displaystyle F(x_{0},x)=\left\{{\begin{array}{ll}{\hbox{var}}_{[x_{0},x]}(f),&x_{0}\leq x\\-{\hbox{var}}_{[x_{0},x]}(f),&x_{0}>x\\\end{array}}\right.\,}
Fixando um
x
0
∈
D
{\displaystyle x_{0}\in D\,}
x
{\displaystyle x\,}
Agora define-se:
p
(
x
)
=
1
2
[
F
(
x
0
,
x
)
+
f
(
x
)
]
q
(
x
)
=
1
2
[
F
(
x
0
,
x
)
−
f
(
x
)
]
{\displaystyle {\begin{array}{l}p(x)={\frac {1}{2}}\left[F(x_{0},x)+f(x)\right]\\~\\q(x)={\frac {1}{2}}\left[F(x_{0},x)-f(x)\right]\end{array}}}
É fácil ver que
f
(
x
)
=
p
(
x
)
−
q
(
x
)
{\displaystyle f(x)=p(x)-q(x)\,}
p
(
x
)
{\displaystyle p(x)\,}
q
(
x
)
{\displaystyle q(x)\,}
y
>
x
{\displaystyle y>x\,}
p
(
y
)
−
p
(
x
)
=
1
2
[
F
(
x
0
,
y
)
+
f
(
y
)
]
−
1
2
[
F
(
x
0
,
x
)
+
f
(
x
)
]
=
1
2
[
F
(
x
0
,
y
)
−
F
(
x
0
,
x
)
]
+
1
2
[
f
(
y
)
−
f
(
x
)
]
=
1
2
[
var
[
y
,
x
]
(
f
)
]
+
1
2
[
f
(
y
)
−
f
(
x
)
]
≥
0
{\displaystyle {\begin{array}{rcl}p(y)-p(x)&=&{\frac {1}{2}}\left[F(x_{0},y)+f(y)\right]-{\frac {1}{2}}\left[F(x_{0},x)+f(x)\right]\\&=&{\frac {1}{2}}\left[F(x_{0},y)-F(x_{0},x)\right]+{\frac {1}{2}}\left[f(y)-f(x)\right]\\&=&{\frac {1}{2}}\left[{\hbox{var}}_{[y,x](f)}\right]+{\frac {1}{2}}\left[f(y)-f(x)\right]\geq 0\end{array}}}
Da penúltipla para a última linha usamos
F
(
x
0
,
y
)
−
F
(
x
0
,
x
)
=
var
[
x
0
,
y
]
(
f
)
−
var
[
x
0
,
x
]
(
f
)
=
var
[
x
,
y
]
(
f
)
{\displaystyle F(x_{0},y)-F(x_{0},x)={\hbox{var}}_{[x_{0},y](f)}-{\hbox{var}}_{[x_{0},x](f)}={\hbox{var}}_{[x,y](f)}}
var
[
x
,
y
]
(
f
)
≥
|
f
(
x
)
−
f
(
y
)
|
{\displaystyle {\hbox{var}}_{[x,y](f)}\geq \left|f(x)-f(y)\right|}
A demontração sendo perfeitamente análoga para a função
q
(
x
)
{\displaystyle q(x)\,}
editar
Considere a função:
f
(
x
)
=
{
x
cos
(
π
x
)
,
x
≠
0
0
,
x
=
0
{\displaystyle f(x)=\left\{{\begin{array}{ll}x\cos \left({\frac {\pi }{x}}\right),&x\neq 0\\0,&x=0\end{array}}\right.}
Esta função não é de variação limitada no intervalo
[
0
,
1
]
{\displaystyle [0,1]\,}
x
n
=
1
n
+
1
,
f
(
x
n
)
=
1
n
+
1
(
−
1
)
n
,
n
=
0
,
1
,
2
,
…
{\displaystyle x_{n}={\frac {1}{n+1}},\quad f(x_{n})={\frac {1}{n+1}}(-1)^{n},n=0,1,2,\ldots \,}
Assim
|
f
(
x
n
)
−
f
(
x
n
−
1
)
|
=
1
n
+
1
+
1
n
>
1
/
n
,
n
=
1
,
2
,
3
,
…
{\displaystyle \left|f(x_{n})-f(x_{n-1})\right|={\frac {1}{n+1}}+{\frac {1}{n}}>1/n,n=1,2,3,\ldots \,}
Portanto,
var
[
0
,
1
]
(
f
)
≥
∑
n
=
1
∞
1
n
=
+
∞
{\displaystyle {\hbox{var}}_{[0,1]}(f)\geq \sum _{n=1}^{\infty }{\frac {1}{n}}=+\infty \,}
![{\displaystyle [a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cb97ebba2cd3175f9a77446963c1849fc353ee)
![{\displaystyle {\hbox{osc}}_{[a,b]}(f):=\sup _{[a,b]}f(x)-\inf _{[a,b]}f(x)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85278f888c11fe6d95ad3480ac22ac67ee089f39)
![{\displaystyle {\hbox{osc}}_{[a,b]}(f)=f(b)-f(a)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c38faf1b0d1cfbcc58dd7454e0a5cf5e61c14fc)
![{\displaystyle {\hbox{osc}}_{[a,b]}(f)=f(a)-f(b)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d49a101a31b5222bb17cad5d437d918f9391d48)
![{\displaystyle D\supset [a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8539811439985d4245b40f79c7e0700351703996)
![{\displaystyle {\hbox{var}}_{[a,b]}(f):=\sup _{P\in \mathbb {P} }{\hbox{var}}_{P}(f)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e89a589050852eca93125845530c282906d13b91)
![{\displaystyle {\hbox{var}}_{[a,b]}(f)=\left|f(a)-f(b)\right|\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d5912f2b75beeb4e2f399b4d75b9c725a2bf9b9)
![{\displaystyle {\hbox{var}}_{[a,b]}(f)\geq {\hbox{var}}_{[c,d]}(f)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0d2a89fae6e748a7d9ac1be459114302eade20)
![{\displaystyle {\hbox{var}}_{[a,b]}(f+g)\leq {\hbox{var}}_{[a,b]}(f)+{\hbox{var}}_{[a,b]}(g)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb439ed35dc828215ef8b64f920bb394ca14f2ad)
![{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)=|\alpha |{\hbox{var}}_{[a,b]}(f),~~\forall ~\alpha \in \mathbb {R} \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcae6bbcf88bb12aebfe4986578ae0fda5e1a416)
![{\displaystyle {\hbox{var}}_{[a,c]}(f)={\hbox{var}}_{[c,b]}(f)+{\hbox{var}}_{[b,c]}(f),~~\forall ~c\in [a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/842edae2475f3840ba5a7eb636504c396c20aa0c)
![{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)\leq \infty \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3a5d9b331b9c3b50d51371fd5fb818aa3882a2)
![{\displaystyle C^{1}[a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6737b2e7f5e20de6e181966092701b300b2c4453)
![{\displaystyle {\hbox{var}}_{[a,b]}(f)=\int _{a}^{b}|f'(x)|dx\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fc8a642ed17c72d7d85af189c4624806df6fbbc)
![{\displaystyle 0\leq {\hbox{var}}_{[a,b]}(f)-{\hbox{var}}_{P_{k}}(f)\leq 1/k\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9a653fac235fc9bdd2bede162eb6a0f89dc4fcb)
![{\displaystyle {\hbox{var}}_{[a,b]}(f)=\lim _{k\to \infty }{\hbox{var}}_{P_{k}}(f)=\lim _{k\to \infty }\sum _{i=1}^{n_{k}}\left|f'(x_{i}^{k}*)\right|(x_{i}^{k}-x_{i-1}^{k})=\int _{a}^{b}|f'(x)|dx\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2c674e8700f3e77bfbfe794d54a42c828c9c3d53)
![{\displaystyle F(x_{0},x)=\left\{{\begin{array}{ll}{\hbox{var}}_{[x_{0},x]}(f),&x_{0}\leq x\\-{\hbox{var}}_{[x_{0},x]}(f),&x_{0}>x\\\end{array}}\right.\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0977bdd557f0246923435333d7c432a9904d7e41)
![{\displaystyle {\begin{array}{l}p(x)={\frac {1}{2}}\left[F(x_{0},x)+f(x)\right]\\~\\q(x)={\frac {1}{2}}\left[F(x_{0},x)-f(x)\right]\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4445602dd885ffbd40eab7bfab9b75a518279f36)
![{\displaystyle {\begin{array}{rcl}p(y)-p(x)&=&{\frac {1}{2}}\left[F(x_{0},y)+f(y)\right]-{\frac {1}{2}}\left[F(x_{0},x)+f(x)\right]\\&=&{\frac {1}{2}}\left[F(x_{0},y)-F(x_{0},x)\right]+{\frac {1}{2}}\left[f(y)-f(x)\right]\\&=&{\frac {1}{2}}\left[{\hbox{var}}_{[y,x](f)}\right]+{\frac {1}{2}}\left[f(y)-f(x)\right]\geq 0\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20e1979e7501e98ccfd46e4f1289cb093d3934eb)
}-{\hbox{var}}_{[x_{0},x](f)}={\hbox{var}}_{[x,y](f)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/af511b6413d713fd5334ca0bc4ff7fe467eb4000)
}\geq \left|f(x)-f(y)\right|}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8aad7adab7d208b55d37e0fd523833e258b76c3d)
![{\displaystyle [0,1]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/43e2b417e116123c724ee6f69cf309f6ad17a2d0)
![{\displaystyle {\hbox{var}}_{[0,1]}(f)\geq \sum _{n=1}^{\infty }{\frac {1}{n}}=+\infty \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a548c747f118ff2709dd41a2246d388528e926)