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E
A E x 7→ d(x, A)
E
K U
α > 0
∀x∈K, B(x, α)⊂U
K E f :K→K
∀(x, y)∈K2, x 6=y=⇒
f(x)−f(y) < x−y
f
c∈K
∀x∈K, 
f(x)−x ≥ f(c)−c
f
K E
f:K→K
∀x, y ∈K, x 6=y=⇒d(f(x), f(y)) < d(x, y)
f
fR R I
f
J
f(J) = I
K E
f:K→K ρ
∀x, y ∈K, kf(y)−f(x)k ≤ ρky−xk
ρ < 1f
ρ= 1 K f
a∈K n ∈N∗
fn:x7→ a
n+n−1 nf(x)
E F
K E f :K→F
L=f(K)L
f−1:L→K
f[0 ; +∞[f
`+∞
f[0 ; +∞[
E f E
x∈E r > 0Bf(x, r)
f(Bf(x, r))
x∈E r 0< r < kxkK=Bf(x, r)
f(K)⊂K
a∈K n ∈N∗
yn=1
n
n−1
X
k=0
fk(a)
(yn)n≥1K f(yn)−yn
0Ew∈K f(w) = w

f(K)⊂K
1∈Sp fSp f⊂[−1; 1]
f
dim E= 3 B= (e1, e2, e3)
E
K=x.e1+y.e2+z.e3|x2
a2+y2 b2+z2 c2≤1a, b, c > 0
f(K) = K−1f

x, x0∈E
∀y∈A, kx−yk≤kx−x0k+kx0−yk
d(x, A)≤ kx−x0k+kx0−ykd(x, A)− kx−x0k ≤ kx0−yk
d(x, A)− kx−x0k ≤ d(x0, A)
d(x, A)−d(x0, A)≤ kx−x0k
|d(x, A)−d(x0, A)|≤kx−x0k
x7→ d(x, A)
x7→ d(x, CEU)K
α= minx∈Kd(x, CEU)x0∈K
α= 0 x0∈ CEUCEU x0/∈U x0∈K
α > 0
∀x∈K, B(x, α)⊂U
f x 6=y
kf(x)−f(y)k<kx−yk
f(x) = x f(y) = y
δ:x7→ kf(x)−xkK
δ K
c∈K
∀x∈K, δ(x)≥δ(c)
f(c)6=c
δ(f(c)) = kf(f(c)) −f(c)k<kf(c)−ck=δ(c)
c f(c) = c
x6=y
d(x, y) = d(f(x), f(y)< d(x, y)
g:x7→ d(x, f(x)) K
g K g
x0∈K
f(x0)6=x0
g(f(x0)) = d(f(f(x0)), f(x0)) < d(f(x0), x0) = g(x0)
x0f(x0) = x0
α, β I
a, b ∈Rα, β
f([a;b]) = [α;β]f[a;b]
A={x∈[a;b]|f(x) = α}B={x∈[a;b]|f(x) = β}
∆ = {|y−x| | x∈A, y ∈B}
∆R
m
(xn)∈AN(yn)∈BN
|yn−xn| → m
A(xn)
(xϕ(n))A(yϕ(n))
B x∞y∞
x∞∈A, y∞∈B|y∞−x∞|= min ∆
I[a;b]
x∞≤y∞J= [x∞;y∞]

α, β ∈f(J)f(J)
I⊂f(J)
γ∈f(J)c∈J f(c) = γ
γ < α [z;y∞]
A y∞x∞
γ > β f(J)⊂I
f
g:x∈K7→ kf(x)−xkg
x0∈K
g(x0)≤g(f(x0)) = kf(f(x0)) −f(x0)k ≤ ρkf(x0)−x0k=ρg(x0)ρ < 1
g(x0) = 0 f(x0) = x0
f
K fnK
K
kfn(y)−fn(x)k=n−1
nkf(y)−f(x)k ≤ ρnky−xk
ρn<1
fnxn(xn)
K(xϕ(n))
x∞∈K
fϕ(n)(xϕ(n)) = xϕ(n)
a
ϕ(n)+ϕ(n)−1 ϕ(n)f(xϕ(n)) = xϕ(n)
f(x∞) = x∞
L L
f−1∃y∈L, ∃ε > 0,∀α > 0,∃y0∈L
|y0−y| ≤ α
f−1(y0)−f−1(y) > ε
x=f−1(y)α=1
nyn∈L xn=f−1(yn)
|yn−y| ≤ 1
n|xn−x|> ε (xn)
K(xϕ(n))
a= lim xϕ(n)f yϕ(n)=f(xϕ(n))→f(a)yn→y
y=f(a)a=f−1(y) = x
xϕ(n)−x > ε
ε > 0A∈R+
∀x≥A, |f(x)−`| ≤ ε/2
∀x, y ∈[A; +∞[,|f(y)−f(x)| ≤ ε
f[0 ; A]α > 0
∀x, y ∈[0 ; A],|y−x| ≤ α=⇒ |f(y)−f(x)| ≤ ε
x, y ∈R+|y−x| ≤ α x ≤y
x, y ∈[0 ; A]|f(y)−f(x)| ≤ ε
x, y ∈[A; +∞[|f(y)−f(x)| ≤ ε
x∈[0 ; A]y∈[A; +∞[|x−A| ≤ α
|f(x)−f(y)| ≤ |f(x)−f(A)|+|f(A)−f(y)| ≤ 2ε
ε
g=f◦tan [0 ; π/2[
π/2
f=g◦arctan arctan
f
Bf(x, r)
f
f(Bf(x, r))
K f(K)f
fk(a)K yn
K

f(yn)−yn=1
n(fn(a)−a)
K(fn(a)−a)n≥1
f(yn)−yn−→
n→+∞0E
(yn)n≥1K
w∈K
yϕ(k)−→
k→+∞w
f(yϕ(k))−yϕ(k)−→
k→+∞0E
f(w) = w
0E/∈K w 6= 0Ef(w) = w
f
λ f v kvk< r
x+v K fn(x+v) = fn(x) + λnv
K(fn(x+v))
(fn(x)) (λnv)|λ| ≤ 1
fR3
A=
100
001
000
f
x= (1,0,0) r= 1/2f(K)⊂K
f(K) = K e1/a e2/b e3/c
f f
λ f v
v v ∈K fn(v) = λn.v ∈K
n∈N|λ| ≤ 1
f−1(K) = K|λ| ≥ 1
|λ|= 1