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U B(0, R)
∀a, b ∈U,
f(b)−f(a)
≤ kb−ak
∀c, d ∈B(0, R),
f−1(d)−f−1(c)
≤ kd−ck
∀a, b ∈U,
f(b)−f(a)
=kb−ak
∀a, b ∈U, (f(a)|f(b)) = (a|b)
f(b)Rn
f U
x0∈Rng:x7→ f(x0+x)−f(x0)
f
dϕ(0) = IdRn
ψ:x7→ ϕ(x)−x
ψC1dψ(0) = ˜
0B
∀x∈B,
dψ(x)
≤1
2
∀x, y ∈B,
ψ(y)−ψ(x)
≤1
2ky−xk
x, y ∈B ϕ(x) = ϕ(y)ψ(y)−ψ(x) = y−x
ky−xk ≤ 1
2ky−xk
y=x
θ= (dϕ)−1(0) ◦ϕC1
dθ(0) = (dϕ−1)(0) ◦dϕ(0) = IdRn
V θ
V
ϕ V
h−f a
h−f a
h−f a f
h a `
u→0
f(a+u) = f(a) + `(u)+o1(u)h(a+u) = h(a) + `(u)+o2(u)
g(a) + `(u)+o1(u)≤g(a+u)≤g(a) + `(u)+o2(u)
g(a+u) = g(a) + `(u) + o(u)
g a a `
C1
r
r2sin θ
ϕ: (u, v)7→ (u+v, uv)C1UR2
(s, p)∈R2
(s, p) = ϕ(u, v)u v x2−sx +p= 0
∆ = s2−4p > 0
ϕ
V=(s, p)∈R2s2−4p > 0
(s, p)∈V(u, v)u<v
ϕ(u, v)=(s, p)x2−sx +p= 0
u=s−ps2−4p
2v=s+ps2−4p
2
ϕ U V
U V
ϕ ϕ−1
C1