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f(x+y) = f(x)+
f(y)
f:R→R
∀(x, y)∈R2, f (x+y) = f(x) + f(y)
(R,+)
f:x7→ ax, a
R Q
(R,+)
Q
f(R,+)
∀a∈R,∀r∈Q, f (ra) = rf (a).
a= 1, λ =f(1) , f (r) = λr r ∈Q.
f(R,+)
f
fR
Q
(ei)i∈IR Q fR
f(ej) = 1 f(ei) = 0 i∈I\ {j}j I,
(47.1)
(R,+)
R

fRf(x+y) =
f(x) + f(y)f(xy) = f(x)f(y)x, y R.
f(1) = (f(1))2, f (1) = 0 f(1) = 1. f (1) = 0,
x∈Rf(x) = f(x)f(1) = 0 f
0.
f f (1) = 1.
f(x2) = (f(x))2≥0, f (x)≥0x≥0x≥y
R, f (x)−f(y) = f(x−y)≥0, f
f(x) = x x ∈Rλ=f(1) = 1
R
f(x∧y) = f(x)∧f(y)
R3.
f(47.1) 0,
R.
0
f(47.1) 0,
f(47.1) [a, b]a < b,
m≤M m ≤f(x)≤M x ∈[a, b].
x∈[0, b −a], x +a∈[a, b], m ≤f(x+a)≤M(47.1) ,
m′=m−f(a)≤f(x)≤M′=M−f(a)
x∈[0, b −a].
n∈N\ {0}x∈0,b−a
n, nx ∈[0, b −a], m′≤f(nx) =
nf (x)≤M′m′
n≤f(x)≤M′ n.
ε > 0, n ≥1m′
n,M′ n
]−ε, ε[x∈0,b−a
n,−ε<f(x)< ε. f
0.
f(47.1)
f(47.1)

f(xy) = f(x) + f(y)R∗
(47.1)
(R,+) (C,+) .
f, g =ℜ(f)f h =ℑ(f)f
(R,+)
f(R,+) (C,+) .
f0
f
α f (x) = αx x.
(47.1) C C,
(C,+)
z=x+iy ∈C, f (z) = f(x) + f(iy).
f0,
x7→ f(x)y7→ f(iy)
α, β f (x) = αx f (iy) = βy x, y.
f(z) = αz+z
2+βz−z 2i=γz +δz,
γ, δ
f0
f(xy) = f(x) + f(y)R∗
f:R→R
∀(x, y)∈R2, f (xy) = f(x) + f(y)
R. f = 0 f(0) =
f(x·0) = f(x) + f(0) f(x) = 0 x.
R∗.
f f (1) = f(12) = 2f(1) , f (1) = 0,0 = f(1) =
f(−1)2= 2f(−1) f(−1) = 0 f(−x) = f(−1) + f(x)f(−x) = f(x)
x∈R, f
∀(x, y)∈R+,∗2, f (xy) = f(x) + f(y)
R+,∗
R+,∗(47.2)
∀x∈R+,∗, f (x) = λx
1
dt
t,
λ

R+,∗
1x7→ 1
x,ln (x) = x
1
dt
tx > 0.
R∗(47.2)
f(x) = λln (|x|)x∈R∗.
ln′(x)>0R+,∗ln R+,∗ln (2) >
ln (1) = 0,ln (2n) = nln (2) lim
x→+∞ln (x) = +∞.
ln 1
x=−ln (x) ln x1 x= 0 lim
x→0ln (x) = −∞.
ln R+,∗R,
x7→ ex.
(x∈Ry=ex)⇔(y > 0x= ln (y)) .
ln′(x)>0,
(ex)′=exx∈R.
ln (xy) = ln (x) + ln (y)x, y R+,∗,
∀(x, y)∈R2, f (x+y) = f(x)f(y).
f(47.2) R+,∗, g Rg(t) = f(et)
f(47.2) R+,∗.
f
f1
fR+,∗
λ f (x) = λln (x)x∈R+,∗.
x7→ xr,
r∈Qx∈]0,+∞[
x > 0,lim
n→+∞(n
√x) = 1.
n≥1, un
∀x∈R+,∗, un(x) = nn
√x−1
vn=un−un+1 ]0,1[ ,
]1,∞[vn(1) = 0.
x∈R+,∗\ {1},(un(x))n∈N∗
(un(x))n≥1λ(x)λ(x)=0
x= 1, λ (x)>0x > 1λ(x) = −λ1
x<0 0 < x < 1.

f(xy) = f(x) + f(y)R∗
x > 0, y > 0n≥1,
un(xy) = un(x) + un(y) + un(x)un(y)
n
λ(47.2) R+,∗.
x > 0,
x−1
x≤λ(x)≤x−1
λ1
λR+,∗1
x1.
n
√1 = 1 n
1
x=1
n
√x, x > 1.
x > 1.
αn(x) = n
√x−1, αn(x)>0n≥1
x= (1 + αn(x))n>1 + nαn(x),
0< αn(x)<x−1
n
lim
n→+∞(αn(x)) = 0,lim n→+∞(n
√x) = 1.
x > 1,(n
√x)n∈N∗
n+1
√x < n √x x < xn+1
n=xn
√xn √x > 1
1, ℓ (x)≥1. ℓ (x)>1, λ ∈]1, ℓ (x)[
n0n
√x > λ n ≥n0, x > λn
n≥n0lim
n→+∞λn= +∞. ℓ (x) = 1.
vnR+,∗x > 0
v′
n(x) = x1
n−1−x1
n+1 −1=x−n−1
n−x−n
n+1 =x−n n+1 x1
n(n+1) −1.
vn]0,1[ ,
]1,∞[v′
n(1) = vn(1) = 0.
vn(x)>0x∈R+,∗\ {1}, un(x)<
un+1 (x) (un(x))n∈N∗x= 1,
(un(1))n∈N∗0.
un(1) = 0, un1
x=n1
n
√x−1=−un(x)
n
√xlim
n→+∞(n
√x) = 1,
x > 1. x > 1.