Cours Équations fonctionnelles Pierre Bornsztein

N Z Q

f:R→Rx y
f(x+y) = f(x) + f(y)
XR
f X Rx X
f(x)x f
X f
x∈Rf(x) = x2R
R
n6= 0 f(n)
Z?R
f(n)
f:X→Rf(X) = {f(x), x ∈X}
f X Y Y =f(X)x∈X
Y y ∈Y f
f X Y y ∈Y
f a, b ∈X f(a) = f(b)a=b
f X Y
X Y
f:x7→ x2R R −1
f(R) = R+fR R+
R R R R+f(1) =
1 = f(−1)
fR+R R−R
fR+R+R−R+)
f:A→B
A B

f:X→Y f
f−1f−1:Y→X
y∈Y f−1(y) = x, x X f(x) = y
x y f
f:x7→ x2
R+f−1:x7→ √x f
R−R+f−1:x7→ −√x
f:X→RI X
f I
a, b ∈I a < b f(a)6f(b)f(a)< f(b)
f I
a, b ∈I a < b f(a)>f(b)f(a)> f(b)
f I I
I
f:X→RX f
XR
If X a, b ∈X
a < b f(a)< f(b)b < a f(b)< f(a)
f(a)6=f(b)f X R
f X J
f:X→Y
X f−1
Y
If a, b ∈Y a < b
x, y ∈X f(x) = a f(y) = b f−1(a) = x f−1(b) = y
a < b f(x)< f(y)f
x < y f−1(a)< f−1(b)f−1
Y
fJ
fR
x∈Rf(f(x)) = x

If(x) = x x ∈R
f a f(a)6=a
f(a)< a f f(f(a)) < f(a)
a < f(a)
f(a)> a f(f(a)) > f(a)a > f(a)
a
f(a) = aJ
f:X→Rf
x∈X(−x)∈X
x∈X f(−x) = f(x)f(−x) = −f(x)
f f(n) = n2n>0
N
f f(n) = n2n
f n f(n) =
n
f:R→R
x, y ∈R
f(2x) = f³sin ³π
2x+π 2y´´+f³sin ³π 2x−π 2y´´
f(x2−y2) = (x+y)f(x−y) + (x−y)f(x+y)
If
x∈Ry= 0
f(2x) = 2f³sin ³π
2x´´
y= 2 x−x)
f(−2x) = f³sin ³−π
2x+π´´+f³sin ³−π 2x−π´´= 2f³sin ³π 2x´´=f(2x)
f
y∈Rx= 0
f(−y2) = yf(−y)−yf(y) = y(f(−y)−f(y)) = 0
fR−f
fR
J

f:X→RT > 0f T
T
x∈X x +T∈X
x∈X f(x) = f(x+T)
a f :R→Rx
f(x+a) = 1
2+pf(x)−(f(x))2
f
a= 1
If
1
26f(x)61x∈R
g:x7→ f(x)−1
2x06g(x)61 2
g(x+a) = r1
4−(g(x))2
(g(x+a))2=1
4−(g(x))2
(g(x+ 2a))2=1
4−(g(x+a))2= (g(x))2
g(x+ 2a) = g(x)f(x+ 2a) = f(x)
f2a
f:x7→ 1
2¯¯sin ¡π 2x¢¯¯+1 2J
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•f(x)f(0) f(1) f(−1) f(2)
f(0) f(1)
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