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f:x7→ y=f(x) [a, b]R R+A
a6x6b
06y6f(x)
n[a, b]n
[a0, a1],[a1, a2],...,[an−1, an]
a0=a, an=bb−a
nk0n
ak=a+kb−a
n
dnDn[ak, ak+1]
mkf[ak, ak+1] (ak+1 −ak)×mkb−a
n×mk
dn=b−a
n
n−1
X
k=0
mk
DnMkf[ak, ak+1]
Dn=b−a
n
n−1
X
k=0
Mk
∀n∈N∗, dn6A6Dn
n

f∈ C([a, b],R+) (dn) (Dn)n+∞
a b f R[a,b]fRb
af(x)x
Z[a,b]
f= lim
n→+∞
b−a
n
n−1
X
k=0
mk= lim
n→+∞
b−a
n
n−1
X
k=0
Mk
∀n∈N∗, dn6A6Dn
A=Z[a,b] f
x∈]0,π
2] [0, x]t7→ cos t dn
Rx
0cos t t 0x
mkMk
f[ak, ak+1]mk6f(ak)6Mkmk6fa+kb−a
n6Mk
dn6b−a
n
n−1
X
k=0
fa+kb−a
n6Dn
n+∞
Z[a,b]
f= lim
n→+∞
b−a
n
n−1
X
k=0
fa+kb−a
n
Z[a,b]
f= lim
n→+∞
b−a
n
n
X
k=0
fa+kb−a
n= lim
n→+∞
b−a
n
n
X
k=1
fa+kb−a
n
f[0,1] a= 0 b= 1
lim
n→+∞
1
n
n−1
X
k=0
fk
n= lim
n→+∞
1
n
n
X
k=1
fk
n=Z1
0
f(t)t
lim
n→∞ 1
n2+ 12+2 n2+ 22+··· +n 2n2
Sn=
n
X
k=1
√k
lim
n→∞
n
r(2n)!
nnn!

[a, b]x7→ f(x)
R+x x1, x2, . . . , xn
f(x1), f(x2), . . . , f(xn)
x1, x2, . . . , xn[a, b]
n x1
[a0, a1]x2[a1, a2]xk[ak−1, ak]
k f(xk)f[ak−1, ak]
mk−16f(xk)6Mk−1
1
n(m0+m1+··· +mn−1)61 n(f(x1) + f(x2) + ··· +f(xn)) 61 n(M0+M1+··· +Mn−1)
dn
b−a61 n(f(x1) + f(x2) + ··· +f(xn)) 6Dn b−a
n(dn) (Dn)Z[a,b] f
lim
n→+∞
1
n(f(x1) + f(x2) + ··· +f(xn)) = 1 b−aZ[a,b]
f
f[a, b]
f[a, b]R+f[a, b]1
b−aZ[a,b]
f
R+
R C
Za
a
f(t)t= 0
a < b < c f ∈ C([a, c],R+)
Z[a,c]
f=Z[a,b] f+Z[b,c] f
a, b, c Z[b,a] f=−Z[a,b] f
f∈ C([a, b],R C)n∈N∗
σn(f) = b−a
n
n−1
X
k=0
fa+kb−a
n
n7→ σn(f)
R+Z[a,b] f= lim
n→+∞σn(f)
f[a, b]
[a, b]Rm
f∈ C([a, b],R C)
Z[a,b]
f= lim
n→+∞σn(f)σn(f) = b−a
n
n−1
X
k=0
fa+kb−a
n

∀x∈[a, b]f(x) = CZ[a,b] f=C(b−a)
fZ[a,b] f= 0
∀α, β ∈C,∀f, g ∈ C([a, b],C),Z[a,b] (αf +βg) = αZ[a,b] f+βZ[a,b] g
f=f1+if2f1, f2∈ C([a, b],R)Z[a,b] f=Z[a,b] f1+iZ[a,b] f2
f=f1+if2, f1=f+
1−f− 1, f2=f+ 2−f− 2
f∈ C([a, b],R+)a<b Z[a,b] f>0
f, g ∈ C([a, b],R)a<b ∀t∈[a, b], f(t)6g(t)
Z[a,b]
f6Z[a,b] g
∀t∈[a, b], g(t)−f(t)>0Z[a,b] (g−f)>0Z[a,b] g−Z[a,b] f>0
lim
n→+∞Z1
0
xnexxlim
n→+∞Z2n2
n2
arctan x
n
xxlim
x→0+Z3x
x
t
tet
gR R g+g−
g+(t) = max(g(t),0) g−(t) = −min(g(t),0)
g=g+−g−|g|=g++g−

f[a, b]R+
Zb
a
f(t)t= 0 =⇒ ∀t∈[a, b], f(t) = 0
t0∈[a, b]f(t0)>0t0∈]a, b[
f(t0) = A f t0I= [t0−δ, t0+δ]∀t∈I, f(t)>A
2
ε=A
2
g:I→R+
t7→ A
2
ZI
g= 2δ×A
2=δA > 0
∀t∈I, f (t)>g(t)Z[a,b] f>ZI f>ZI gZ[a,b] f > 0
f∈ C([a, b],C)a<b
Zb
a
f(t)t
6Zb
a|f(t)|t6(b−a) sup
t∈[a,b]|f(t)|
∀n, |σn(f)|6σn(|f|)n→+∞σn
f[a, b]RfZb
a
f(t)t= (b−a) sup
t∈[a,b]|f(t)|
f:I→C R2Rm∀t∈I, f0(t)=0 f I
f∈ C(I, C)a∈I F :
I→C
x7→ Zx
a
f(t)tC1
f I a
∀x∈I, F 0(x) = f(x)