# Continuité et équation fonctionnelle

f:RR
xR, f(x) = f(x2)
f
f:RRxR
fx+ 1
2=f(x)
f
f:RR
xR, f(2x) = f(x) cos x
f
f:RR
x, y R, f(x+y) = f(x) + f(y)
f(0) xRf(x) = f(x)
nZxRf(nx) = nf(x)
rQf(r) = ar a =f(1)
xRf(x) = ax
f:RRx, y R
f(x+y) = f(x) + f(y)
f x0R
f
f:RR
x, y R, f x+y
2=1
2(f(x) + f(y))
f f(0) = f(1) = 0
f
xR,2f(x) = f(2x)
f
f
f:RR
x, y R, f x+y
2=1
2(f(x) + f(y))
f(0) = 0
x, y R, f (x+y) = f(x) + f(y)
f
xR, f(x) = f((x)2) = f(x2) = f(x)
f
x > 0x1/2n
n→∞ 1f(x1/2n)
n→∞ f(1) f
f(x1/2n) = f(x1/2n1) = · · · =f(x)
f(x) = f(1) x > 0xR
f(0) = limx0+f(x) = f(1)
xR, f(x) = f(1)
xR(un)u0=x n N
un+1 =un+ 1
2
x1 (un)
x1 (un)
(un)
nNf(x) = f(un)f(x) = f(1)
f
f(x) = fx
2cos x
2=fx
4cos x
4cos x
2=. . . =fx
2ncos x
2n. . . cos x
2
sin x
2ncos x
2n. . . cos x
2=1
2nsin x
sin x
2nf(x) = sin x
2nfx
2n
x6= 0 n+sin x
2n6= 0
f(x) = sin x
2nsin x
2nfx
2nsin x
xf(0)
xR, f(x) = sin x
x
x=y= 0 f(0) = 2f(0) f(0) = 0
y=x f(0) = f(x) + f(x)f(x) = f(x)
nNf(nx) = nf(x)
nZn=p p N
f(nx) = f(px) = pf(x) = nf(x)
rQr=p/q p ZqN
f(r) = pf(1/q) = p
qqf(1/q) = p
qf(1) = ar
xR(un)unx unQ
f(un)f(x)unQf(un) = aunax
f(x) = ax
f(x+y) = f(x) + f(y)
rQ, f(r) = rf(1)
aR,nZ, f(na) = nf(a)
n= 0
f(0) = f(0) + f(0)
f(0) = 0
nN
f((n+ 1)a) = f(na) + f(a)
nZ
f(x) = f(x)
f(x) + f(x) = f(0) = 0
r=p/q QpZqN
f(r) = fp×1
q=pf 1
qf(1) = fq×1
q=qf 1
q
f(r) = p
qf(1) = rf(1)
xR
xR(xn)QNxnx
xn+x0xx0f x0
f(xn+x0x)f(x0)
f(xn+x0x) = f(x0)+(f(xn)f(x))
f(xn)f(x)0
f(x) = lim
n+f(xn) = xf(1)
f
f(2 x) + f(x)=0 f(x) + f(x)=0 f(x) = f(x+ 2) f
f(x/2) = f(x)/2f(2x) = 2f(x)
f f
f(2x) = 2f(x)f
f
a=f(1) f(0) b=f(0) g(x) = f(x)(ax +b)
g f
xR, f x
2=fx+ 0
2=1
2(f(x) + f(0)) = 1
2f(x)
x, y R, f x+y
2=1
2f(x+y)
x, y R, f (x+y) = f(x) + f(y)
f
f x 7→ ax
x7→ f(x)f(0)
f x 7→ ax +b
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# Continuité et équation fonctionnelle

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