Fonctions à variation bornée

f: [a, b]→Rσ= (a=x0<
x1< . . . <n=b) [a, b]f σ
V(f, σ) =
n−1
X
k=0
|f(xk+1 −f(xk)|
Vb
a(f) = sup
σ
V(f, σ)∈[0,+∞]
f[a, b]Vb
a(f)f
[a, b]
fC1[a, b]f[a, b]
f
Vb
a(f+g)Vb a(f) + Vb a(g)
c∈]a, b[f[a, b]
[a, c] [c, b]
f[a, b]f
[a, b]
σ= (a=x0< x1< . . . < xn=b) [a, b]k∈ {0, . . . n −1}
|f(xk+1)−f(xk)|=¯
¯
¯
¯Zxk+1
xk
f0(t)dt¯
¯
¯
¯
≤Zxk+1
xk
|f0(t)|dt
V(f, σ)≤Zb
a
|f0(t)|dt f [a, b]Vb
a(f)≤
Zb
a
|f0(t)|dt
Vb
a(f)Zb
a
|f0(t)|dt σ [a, b]
k∈ {0, . . . , n −1}ξk∈[xk, xk+1]
f(xk+1)−f(xk)=(xk+1 −xk)f(ξk)V(f, σ) =
n−1
X
k=0
(xk+1 −xk)|f0(ξk)|
0V(f, σ)
Zb
a
|f0(t)|dt |f0|ε > 0σ
V(f, σ)≥Zb
a
|f0(t)|dt −ε V b
a(f) = Zb
a
|f0(t)|dt
f α σ [a, b]
V(f, σ)≤
n−1
X
k=0
α(xk+1 −xk) = α(b−a)f[a, b]Vb
a(f)≤α(b−a)
f[a, b]Vb
a(f)≤α(b−a)
f[a, b]σ[a, b]V(f, σ) =
n−1
X
k=0
(f(xk+1)−f(xk)) =
f(b)−f(a)f[a, b]Vb
a(f) = f(b)−f(a)
Vb
a(f) = f(a)−f(b)

f g [a, b]Vb
a(f+g)≤Vb a(f) + Vb a(g)
Vb
a(f)Vb a(g)f g
σ[a, b]
V(f+g, σ) =
n−1
X
k=0
|(f+g)(xk+1)−(f+g)(xk)|
≤
n−1
X
k=0
|f(xk+1)−f(xk)|+
n−1
X
k=0
|g(xk+1)−g(xk)|
≤V(f, σ) + V(g, σ)≤Vb
a(f) + Vb a(g)
f+g V b
a(f+g)≤Vb a(f) + Vb a(g)
σ[a, b]σ0= (a=x0< x1< . . . < xn=b) [a, b]
σ c p xp=c;σ1= (a=x0< x1<
. . . < xn=b) [a, c]σ2= (xp< . . . < xn=b) [c, b]
V(f, σ)≤V(f, σ0)
V(f, σ0) = V(f, σ1) + V(f, σ2)
V(f, σ)≤V(f, σ1) + V(f, σ2)≤Vc
a(f) + Vb c(f)
Vb
a(f)≤Vc a(f) + Vb c(f)f
[a, c] [b, c]ε > 0σ1σ2[a, c] [c, b]
V(f, σ1)≥Vc
a(f)−ε
2V(f, σ2)≥Vb
c(f)−ε
2
σ[a, b]σ1σ2V(f, σ) = V(f, σ1) + V(f, σ2)≥
Vc
a(f) + Vb c(f)−ε V b a(f) = Vc a(f) + Vb c(f)
Vc
a(f)Vb c(f)Vb a(f)σ1[a, c]
σ2[b, c]σ V (f, σ) = V(f, σ1)+V(f, σ2)
f[a, b]x∈[a, b]
f[a, x]g[a, b]g(x) = Vx
a(f)
a≤x≤x0≤b g(x0) = g(x) + Vx0
x(f)≥g(x)g
Vx0
x(f)≥ |f(x0)−f(x)|σ= (x < x0)
g(x0)≥g(x) + f(x0)−f(x)f(x0)−g(x0)≤f(x)−g(x)h=f−g
f=g+h