Em análise matemática , define-se a variação total de uma função
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
[
a
,
b
]
⊆
D
{\displaystyle [a,b]\subseteq D\,}
var
[
a
,
b
]
(
f
)
=
sup
∑
i
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
{\displaystyle {\hbox{var}}_{[a,b]}(f)=\sup \sum _{i}\left|f(x_{i})-f(x_{i-1})\right|\,}
As variações positiva e negativa de uma função
f
:
D
→
R
{\displaystyle f:D\to \mathbb {R} \,}
[
a
,
b
]
⊆
D
{\displaystyle [a,b]\subseteq D\,}
var
[
a
,
b
]
+
(
f
)
=
sup
∑
(
+
)
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
{\displaystyle {\hbox{var}}_{[a,b]}^{+}(f)=\sup \sum _{(+)}\left|f(x_{i})-f(x_{i-1})\right|\,}
var
[
a
,
b
]
−
(
f
)
=
sup
∑
(
−
)
−
|
f
(
x
i
)
−
f
(
x
i
−
1
)
|
{\displaystyle {\hbox{var}}_{[a,b]}^{-}(f)=\sup \sum _{(-)}-\left|f(x_{i})-f(x_{i-1})\right|\,}
Em todos os casos o supremo é tomado sob todas as possíveis partições
x
1
,
x
2
,
x
3
,
…
{\displaystyle x_{1},x_{2},x_{3},\ldots \,}
[
a
,
b
]
{\displaystyle [a,b]\,}
(
+
)
{\displaystyle (+)}
i
{\displaystyle i}
f
(
x
i
)
≥
f
(
x
i
−
1
)
{\displaystyle f(x_{i})\geq f(x_{i-1})}
(
−
)
{\displaystyle (-)}
i
{\displaystyle i}
f
(
x
i
)
≤
f
(
x
i
−
1
)
{\displaystyle f(x_{i})\leq f(x_{i-1})}
editar
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
1.
var
[
a
,
b
]
(
f
)
=
var
[
a
,
b
]
+
(
f
)
+
var
[
a
,
b
]
−
(
f
)
{\displaystyle {\hbox{var}}_{[a,b]}(f)={\hbox{var}}_{[a,b]}^{+}(f)+{\hbox{var}}_{[a,b]}^{-}(f)}
2.
f
(
x
)
−
f
(
a
)
=
var
[
a
,
b
]
+
(
f
)
−
var
[
a
,
b
]
−
(
f
)
{\displaystyle f(x)-f(a)={\hbox{var}}_{[a,b]}^{+}(f)-{\hbox{var}}_{[a,b]}^{-}(f)}
editar
Uma função
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
f
{\displaystyle f}
Se
f
(
x
)
=
f
1
(
x
)
−
f
2
(
x
)
{\displaystyle f(x)=f_{1}(x)-f_{2}(x)}
f
1
,
f
2
{\displaystyle f_{1},f_{2}}
var
[
a
,
b
]
(
α
f
)
=
var
[
a
,
b
]
(
α
(
f
1
(
x
)
−
f
2
(
x
)
)
)
≤
var
[
a
,
b
]
(
α
f
1
(
x
)
)
−
var
[
a
,
b
]
(
α
f
2
(
x
)
)
∞
{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)={\hbox{var}}_{[a,b]}(\alpha (f_{1}(x)-f_{2}(x)))\leq {\hbox{var}}_{[a,b]}(\alpha f_{1}(x))-{\hbox{var}}_{[a,b]}(\alpha f_{2}(x))\infty }
Por outro lado, se
f
{\displaystyle f}
[
a
,
b
]
{\displaystyle [a,b]}
f
1
(
x
)
=
var
[
a
,
b
]
+
(
f
)
{\displaystyle f_{1}(x)={\hbox{var}}_{[a,b]}^{+}(f)}
f
2
(
x
)
=
var
[
a
,
b
]
−
(
f
)
{\displaystyle f_{2}(x)={\hbox{var}}_{[a,b]}^{-}(f)}
f
1
{\displaystyle f_{1}}
f
2
{\displaystyle f_{2}}
f
(
x
)
=
var
[
a
,
b
]
+
(
f
)
+
f
(
a
)
−
var
[
a
,
b
]
−
(
f
)
=
f
1
(
x
)
+
f
2
(
x
)
{\displaystyle f(x)={\hbox{var}}_{[a,b]}^{+}(f)+f(a)-{\hbox{var}}_{[a,b]}^{-}(f)=f_{1}(x)+f_{2}(x)}
editar
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\,}
A continuidade , no entanto, não garante que a função seja de variação limitada, um contra-exemplo é:
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 é contínua mas 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 \,}
editar
Stein, Elias M.; Shakarchi, Rami (2007), Real Analysis , Princeton University 
![{\displaystyle [a,b]\subseteq D\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2deb05706b94f18df19c72f20260aad03719930c)
![{\displaystyle {\hbox{var}}_{[a,b]}(f)=\sup \sum _{i}\left|f(x_{i})-f(x_{i-1})\right|\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/596c9b6f8062f9659e9fc404eeac91c5da72f802)
![{\displaystyle {\hbox{var}}_{[a,b]}^{+}(f)=\sup \sum _{(+)}\left|f(x_{i})-f(x_{i-1})\right|\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d554dcae28eb9014ef4b684bcf02f0b06bc43d73)
![{\displaystyle {\hbox{var}}_{[a,b]}^{-}(f)=\sup \sum _{(-)}-\left|f(x_{i})-f(x_{i-1})\right|\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75e778a004c9e308b7c3f49ba9f874bed764d8e2)
![{\displaystyle [a,b]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23cb97ebba2cd3175f9a77446963c1849fc353ee)
![{\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]}(f)={\hbox{var}}_{[a,b]}^{+}(f)+{\hbox{var}}_{[a,b]}^{-}(f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ac3b61e0a3e01d9df67e669791f2eb1e6bc991e)
![{\displaystyle f(x)-f(a)={\hbox{var}}_{[a,b]}^{+}(f)-{\hbox{var}}_{[a,b]}^{-}(f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59b99c0799cfd3a539a7b23109d718e626ee5a4d)
![{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)<\infty \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78117bb7386a2ed36f9eede6c0056d2e7da6e4db)
![{\displaystyle [a,b]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c4b788fc5c637e26ee98b45f89a5c08c85f7935)
![{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)=\alpha \sup \sum _{i}\left|f(x_{i})-f(x_{i-1})\right|=\alpha \sup \sum _{i}f(x_{i})-f(x_{i-1})\leq \alpha (f(b)-f(a))<\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cb42b0adf8c1c7f7c422b00fdebd118b71305cd)
![{\displaystyle {\hbox{var}}_{[a,b]}(\alpha f)={\hbox{var}}_{[a,b]}(\alpha (f_{1}(x)-f_{2}(x)))\leq {\hbox{var}}_{[a,b]}(\alpha f_{1}(x))-{\hbox{var}}_{[a,b]}(\alpha f_{2}(x))\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ff1d5899e2be577fecb7dd0327c47bb574d40c)
![{\displaystyle f_{1}(x)={\hbox{var}}_{[a,b]}^{+}(f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10244337cb19e8ab249840e28aa24f11d72e2d63)
![{\displaystyle f_{2}(x)={\hbox{var}}_{[a,b]}^{-}(f)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b3f85e74b394006fd3be027dd18cfb777bb0bc)
![{\displaystyle f(x)={\hbox{var}}_{[a,b]}^{+}(f)+f(a)-{\hbox{var}}_{[a,b]}^{-}(f)=f_{1}(x)+f_{2}(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f5931c68992c900896e4237951f37fc278b2646)
![{\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 [0,1]\,}](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)
