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f:R→R
f(x) = x
1 + |x|
fR]−1 ; 1[
y∈]−1 ; 1[ f−1(y)
f(x)
a<b∈Rf: ]a;b[→R
f f(]a;b[) = ]limaf; limbf[
α1/e
f∈ C1(R,R)
∀x∈R, f0(x) = αf(x+ 1)
α= 1/e
f: [0 ; +∞[→[0 ; +∞[
f◦f= Id
f

[0 ; +∞[
f(x) = x
1 + x= 1 −1 1 + x
f(0) = 0 lim+∞f= 1
f[0 ; +∞[ [0 ; 1[
]−∞ ; 0[
f(x) = x
1−x=−1 + 1 1−x
lim0f= 0 lim−∞ f=−1
f]−∞ ; 0[ ]−1 ; 0[
fR]−1 ; 1[
y∈[0 ; 1[ x=f−1(y) [0 ; +∞[
y=f(x)⇐⇒ y=x
1 + x⇐⇒ x=y 1−y
y∈]−1 ; 0[ x=f−1(y) ]−∞ ; 0[
y=f(x)⇐⇒ y=x
1−x⇐⇒ x=y 1 + y
∀y∈]−1 ; 1[, f−1(y) = y
1− |y|
limaflimbf f
( =⇒)f
f f ]a;b[
]limaf; limbf[
(⇐= ) f(]a;b[) = ]limaf; limbf[
x0∈]a;b[ limaf < f(x0)<limbf
ε > 0y+∈]f(x0) ; f(x0) + ε]∩]limaf; limbf[x+∈]a;b[
f(x+) = y+
y−∈[f(x0)−ε;f(x0)[ ∩]limaf; limbf[x−∈]a;b[
f(x−) = y−
f x−< x0< x+α= min(x+−x0, x0−x−)>0
x∈]a;b[|x−x0| ≤ α x−≤x≤x+y−≤f(x)≤y+
|f(x)−f(x0)| ≤ ε
f x0f]a;b[
f
f(x)=eβx
α=βe−βf
β7→ βe−β
α∈[0 ; 1/e] β∈R+βe−β=α
f
α= 1/ex7→ exx7→ xex
α∈]0 ; 1/e[
βe−β=α
f
[0 ; +∞[
x∈[0 ; 1] f(x)< x
f(f(x)) < f(x)
f(f(x)) < x f ◦f= Id f(x)> x
f= Id