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f: [a, b]→RC1
L(f) = Zb
ap1+(f0(t))2dt
f
L(f)f[0,1] f(t) = t
L(f)f[0,1] f(t) = ch(t)
L(f)f[0,1/√2] f(t) = √1−t2
f[0,1] f(t) = t2L(f)
f
[1/2,1] f(t)=1/t
L(f)
L(f)
L(f)
f[1,2]
L(f)
α∈R\N
u7→ (1 + u)α
t∈]0,1[
√1 + t4
t2=1 t2+
+∞
X
n=1
(−1)n−1(2n)!
(2n−1)22n(n!)2t4n−2
n an=(2n)!
(2n−1)22n(n!)2
(an)n∈N∗ann
+∞L(f)
L(f) 5
n≥1pn
[0,1]
∀t∈[0,1], pn(t) = tn
(λn)n∈N∗
∀n∈N∗, λn=L(pn) = Z1
0p1 + n2t2n−2dt
(λn)n∈N∗
λ1λ2
pnn
(λn)n∈N∗
(λn)n∈N∗
n∈N∗
λn−nZ1
0
tn−1dt =µnµn=Z1
0
dt
√1 + n2t2n−2+ntn−1
λn<2n∈N∗
(µn)n∈N∗
(λn)n∈N∗
f: [0,1] →R
C1f(0) = 0 f(1) = 1 L(f)<2

R1
0
sin(t)
tdt
R1
0
sin(t)
tdt
x≥1
Zx
1
sin(t)
tdt = cos(1) −cos(x) x−Zx
1
cos(t)
t2dt
R+∞
1
sin(t)
tdt
R+∞
1
cos(2t)
tdt
R+∞
1
sin2(t)
tdt
R+∞
1
|sin(t)|
tdt
g]0,1] g(t) = 1
tsin 1 t
f f(x) = R1
xg(t)dt
f0
f
f[0,1]
]0,1]
lim
x→0+Z1
x|g(t)|dt = +∞
x∈]0,1] λ(x)
f[x, 1]
λ(x)x∈]0,1]
lim
x→0+λ(x) = +∞
f7→ kfk∞= sup
t∈[0,1] |f(t)|
E=C0([0,1],R) [0,1]
R
E1=C1([0,1],R)
[0,1] Rf∈E1
kfk=|f(0)|+kf0k∞
k.k k.k∞
f7→ kfkE1
∀f∈E1,kfk∞≤ kfk
k.k k.k∞E1
(fn)n∈N∗[0,1]
∀n∈N∗,∀t∈[0,1], fn(t) = sin(nπt)
√n
(fn)
[0,1]
n∈N∗In=L(fn)
fn
∀n∈N∗, In≥√nπ
2
L:f7→ L(f) (E1,k.k∞)
L:f7→ L(f) (E1,k.k)