228 - Continuité et dérivabilité des fonctions réelle d`une variable

DRf:D→R
a∈D f a f(x)→x→a, x∈Df(a)
∀ε > 0,∃η > 0 : |x−a|< η =⇒ |f(x)−f(a)|< ε
f D D
f a ∈D∀(xn)∈DN, xn→a
f(xn)→f(a)
R
1Q
C0(D, R)D
R
C0(D, R)R
f∈C0(D, R)g∈C0(f(D),R)
g◦f D
1/f D \ {x∈D|f(x)=0}
f:R∗→R;x7→ 1/x R∗
f:D→RD\{a} ∃limx→af(x) =
l˜ f:D→Rf a l a D
f a
f:R∗→R, x 7→ sin(x)/x
f(0) = 1
f:D→RD∀ε >
0,∃η > 0 : ∀x, y ∈D, |x−y|< η =⇒ |f(x)−f(y)|< ε

x7→ x2
x7→
√x
DR
f:D→Rf a ∈Df(x)−f(a)
x−a
f0(a)f D
D
R
D D
x7→ √x
DR
(f+g)0=f0+g0(λf)0=λf0(fg)0=f0g+fg0
a∈D f(a)6= 0 f a 1/f a
(1/f)0(a) = −f0(a)/f(a)2
f:D→Rg:f(D)→Ra f(a)g◦f
a(g◦f)0(a) = f0(a)g0(f(a))
f:D→f(D)D f(D)f−1
f(D) (f−1)0(x)=1/f0(f−1(x)) x
n f :D→R
f(n+1) = (f(n))0f:D→R
Cnn D n
C1
x7→ x2sin(1/x)
0
f g n x
fg
(fg)(n)(x) =
n
X
k=0 n
kf(k)(x)g(n−k)(x)
f
[a, b]F: [a, b]→R;x7→ Rx
af(t)dt F
f

f∈Cn+1(I, R)
a, b ∈I
f(b) =
n
X
k=0
(b−a)k
k!f(k)(a) + Zb
a
(b−t)n
n!f(n+1)(t)dt
f∈Cn(I, R)a, b ∈I
f(b) =
n
X
k=0
(b−a)k
k!f(k)(a) + o(b−a)
f:I→RIRf
I
f:I→Rf I
f(I)
f:I→R
IRf:I→
Rf(I)R
IRf:I→R
f(I)⊂I I ⊂f(I)f
f: [0,1] →[0,1]
f:I→Rf
f[a, b]
]a, b[f(a) = f(b)c∈]a, b[f0(c)=0
f[a, b]
]a, b[c∈]a, b[f0(c)(b−a) = f(b)−f(a)

f[a, b] ]a, b[
f[a, b]f0≥0f0≤0
]a, b[
f[a, b]f0= 0 ]a, b[
f:I→RI
f0(I)
f[a, b]
]a, b[|f0| ≤ M]a, b[|f(b)−f(a)| ≤ M|b−a|
f: [a, b[→R]a, b[
f0(x)→x→al f a f0(a) = l
f:x7→ exp(1/x2)C∞R
(an)n∈N
f∈C∞(R)f(n)(0) = an∀n
f∈C∞([a, b],R)a b R
f a b
f C∞
c(R,R)
f∈Dn+1([a, b],R)
c∈[a, b]
f(b) =
n
X
k=0
(b−a)k
k!f(k)(a) + (b−a)n+1 (n+ 1)! f(n+1)(c)
fn:I→RI fn
f f
fn:I→RC1I fn
f f0
ng f C1I f0=g(fn)
f
Ck
IR(fn)n
C0(I, R) (fn)
f
(fn)

IRS⊂C0(I, R)
∀x, y ∈I, x 6=y, ∃f∈S:f(x)6=f(y)
KRS
C0(I, R)S
S
C(X, R)
Gδ
C0([0,1],R)
Gδ
f:R→R, x 7→ 0x /∈Q1
qx=p/q
R\Q Q
f:R→R
f
f GδR