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Rn
Rn
Rn
Rm
Rm
Rn
Rn
C1UR
C1
ϕ:t7→ f(x(t), y(t))
ψ: (u, v)7→ f(x(u, v), y(u, v), z(u, v))
n3
Rn
Rn
Rn
∀x= (x1, x2, . . . , xn)∈Rn,∀y= (y1, y2, . . . , yn)∈Rn,(x|y) =
n
X
i=1
xiyi
Rn

Rn
x= (x1, x2, . . . , xn)7→ kxk=v
u
u
t
n
X
i=1
x2
iRn
dRn×RnR+
d(x, y) = v
u
u
t
n
X
i=1
(xi−yi)2
Rm
ERnk.k
dRn
a∈E r > 0
a r
B(a, r) = x∈E / kx−ak< r=x∈E / d(a, x)< r
a r
Bf(a, r) = x∈E / kx−ak6r=x∈E / d(a, x)6r
0 1
R2R3R
B(a, r) = x∈R/|x−a|< r=]a−r, a +r[
Bf(a, r) = x∈R/|x−a|6r= [a−r, a +r]
Rm
A⊂RnARnRnA
∀x∈A, ∃r > 0B(x, r)⊂A
]0,1[ R[0,1[ [0,1] R
RnRn
y > 0R2
x > 0R2
∅RnRn
RnRn
RnRn

• ∅ RnRnRnx
Rnr > 0x r Rn
•(Ai)i∈IRn
x∈Si∈IAii0∈I x ∈Ai0Ai0r > 0B(x, r)⊂Ai0
B(x, r)⊂Si∈IAix∈Si∈IAir > 0B(x, r)⊂Si∈IAi
Si∈IAiRn
•(A1,...,Ap)Rn
x∈T16i6pAii x AiAiri>0B(x, ri)⊂Ai
r= min16i6prii p B(x, r)⊂B(x, ri)⊂AiB(x, r)⊂T16i6pAi
T16i6pAiRn
\
p∈N∗−1
p,1 p={0}
A⊂RnARnRnRn−ARn
[0,1] R[0,1[ R
RnRn
x>0R2
N R
∅RnRn
RnRn
RnRn
E A E A E −A E\A
∅A=E E A =∅
E−A=x∈E / x /∈A
(Ai)i∈IE
E−\
i∈I
Ai=[
i∈I
(E−Ai)E−[
i∈I
Ai=\
i∈I
(E−Ai)

ARna∈Rn
A˚ A A
a∈˚ A⇐⇒ ∃r > 0, B(a, r)⊂A
A¯ A A
a∈¯ A⇐⇒ ∀r > 0, B(a, r)∩A6=∅
A A A
(A) = ¯ A\˚ A
A A
(A) =
◦
(Ac)
¯
A= (A)c
˚
A⊂A⊂¯ A
˚
A¯ A(A)
˚
A A ¯ A A
A⇐⇒ A=˚ A
A⇐⇒ A=¯ A
A˚ A¯ A(A)
A=]0,1]
A={(x, y)∈R2, x >0, y >0, x +y < 1}
A={(x, y, z)∈R3, x2+y2+z2<1}
Rn
ARnr > 0A⊂B(0, r)∀x∈A, kxk< r

A⊂B B =⇒A
∅
Rn
Rn0
A⊂Rna∈Rn
A⇐⇒ ∃r > 0, A ⊂B(a, r)
E=RnA E f A Ra= (a1, . . . , an)∈¯ A
f ` a
∀ε > 0,∃η > 0,∀x∈A, hkx−ak6η=⇒ |f(x)−`|6εi
E=RnA E f A Ra= (a1, . . . , an)∈A
f a lim
x→af(x) = f(a)
∀ε > 0,∃η > 0,∀x∈A, hkx−ak6η=⇒ |f(x)−f(a)|6εi
f A f a ∈A
k > 0f:E→Rk
∀x, y ∈E, |f(x)−f(y)|6kkx−yk
f E E
E→R, x 7→ kxk
pi:Rn→R, x = (x1, . . . , xn)7→ xi