[PDF] [PDF] Chapitre8 : Fonctions continues - Melusine

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D R

f:DÑR x0PD f x0ðñf x0( f(x0)) f(x)ÝÝÝÝÑxÑx0f(x0) ðñ @εą0,Dαą0,@xPD,(|x´x0| ăαùñ |f(x)´f(x0)| ăε) @x0PD,@εą0,Dαą0,@xPD,(|x´x0| ăαùñ |f(x)´f(x0)| ăε) f:DÑR D1ĂD f D1f|D1 f D D1 f Dðñ @x0PD ,@εą0,Dαą0,@xPD ,(|x´x0| ăαùñ |f(x)´f(x0)| ăε) f D1ðñ @x0PD1 ,@εą0,Dαą0,@xPD ,(|x´x0| ăαùñ |f(x)´f(x0)| ăε) f D1ðñ @x0PD1 ,@εą0,Dαą0,@xPD1 ,(|x´x0| ăαùñ |f(x)´f(x0)| ăε) 0 1 2 f |[0,1] f [0,1] f Af Bùñf AYB f [a,b] [b,c]aăbăc f [a,c]f [a,c] x0P[a,b[ f x0 [a,b] x0 f x0 f(x0)x0 f f(x0)x0 bf f b x0P]b,c] ā f:DÑR 1 f f(x)ÝÝÝÝÑxÑx0f(x0) f(x0)‰0 1 f(x)ÝÝÝÝÑxÑx01 f(x0) x0PD 1 f D f:DÑRg:EÑR f(D)ĂE g˝f D

x0PD f(x)ÝÝÝÝÑxÑx0f(x0) g(u)ÝÝÝÝÝÝÑuÑf(x0)g(f(x0)) f(x0)PEg f(x0)

f D |f| D f+f´ D f+@xPD,f+(x) =(f(x),0) =1 2 (f(x) +|f(x)|) f+ f´@xPD,f´(x) =(´f(x),0) =1 2 (´f(x) +|f(x)|) f+ xÞÑxα

C0(I,R) C0I ĿC ŀ

f: [a,b]ÑRaăb cP[a,b] d=f(c)

I R f I f(I)

α,βPf(I) dαβ f(I)

dαβ aPIbPI [a,b]ĂII f I f [a,b] dPf(I) f(I) f(I) f(I) R f: [a,b]ÑR f([a,b]) [a,b] a,bPR aăb f: [a,b]ÑR df(a)f(b) g: [a,b]ÝÑR xÞÝÑf(x)´d f(a)ěf(b) g: [a,b]ÝÑR xÞÝÑd´f(x) ā (an)(bn) a

0=ab0=b

nPN g(an+bn 2 2 bn+1=bn an+1=anbn+1=an+bn 2 (an) (bn) @nPN,bn´an=b´a 2 n (an)(bn) ā c g c f(a) f(b) I f(a) f(b) f: [a,b]ÑR aăb J=f([a,b])

J [a,b]

J

α,βP¯R α,βPJ

J

βP¯R(J)

nPN xnP[a,b] yn=f(xn) (xn)nPN (x1n)nPN= (xφ(n))nPN

β=f(l) βPJ

āαPJ

J R f [a,b] f f(x) =1 x ]0,1[ f(]0,1[) =]1,+8[ f(x) =1 x [1,+8[ f([1,+8[) =]0,1] f:xÑxR f(R) = [´1,1] f:xÑx2]´1,1[ f(]´1n1[) = [0,1[ [3π,4π] f I f I J abaăb ¯RI

J¯Rα,β

aPIα=f(a)αPJ

α=afαRJ

f´1:JÑI J

ā ĕ f αββα

f aPI αPJα=f(a) aRI

´8=αf

x

I x0

a

ā b

f´1 u,u1PJ uău1 f´1(u)ăf´1(u1) f´1(u)ěf´1(u1) f f(f´1(u))ě f(f´1(u1))uěu1 f´1 J f ´1(u0) =x0 @WPV(x0),DUPV(u0),@uPUXJ,f´1(u)PW

WPV(x0)

x u1=f(x1)u2=f(x2) u

1ăf(x0)ău2 u1ău0ău2 [u1,u2] u0 J

u

1,u2PJ U= [u1,u2] @uPUXJ,f´1(u)PW

f u0PJ x0 I x0=(I) WXI

U= [u1,+8[ u0

@uPUXJ,f´1(u)PW f´1(u)PIx0=(I) f´1(u)P[x1,x2]ĂW f´1 u0 f´1 J f ĕ ´f

´f I J

f xP ´JyPI y=f´1(x)ðñf(y) =xðñ ´f(y) =´xðñy= (´f)´1(´x) x 04+8 f 1(x) f (x) +8 @Rf(4) +8 f(4) = 4(1´4)ă04ąe f ]0,4] 0f= +8 f ]0,4]]f(4),+8] f(4)ă00P]f(4),+8] ]0,4]

āf [4,+8[ x1P[4,+8[

f(x1) = 0 f(4)‰0xP]0,4[x1P]4,+8[ x‰x1

D R f:DÑR f D

@εą0,Dαą0,@xPD,@x1PD,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) f:DÑR f D f D f D ðñ @εą0,Dαą0,@xPD,@x1PD,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) f D ðñ @xPD,@εą0,Dαą0,@x1PD,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) xPD,εą0

αą0

@u,u1PD,(|u´u1| ăαùñ |f(u)´f(u1)| ăε) u=x u1x1 @x1PD,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) @xPD,@εą0,Dαą0,@x1PD,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) αą0,Dx,x1PR+,(|x´x1| ăα|f(x)´f(x1)| ěε)

ε= 1

n x=nx1=n+1 n |x´x1|=1 n

ăα |x2´x12|=1

n (2n+1 n ) = 2 +1 n

2ěε ā

ε= 2

f D f D

εą0 α=ε

k xÞÑ?

K= [a,b] R aăb

f:KÑR @εą0,Dαą0,@x,x1PK,(|x´x1| ăαùñ |f(x)´f(x1)| ăε) f(x1)| ěε) n |f(xn)´f(x1n)| ěε) 0 x1φ(n)=xφ(n)loomoon

Ñ0 l+8

f l f(xφ(n))Ñf(l)f(x1φ(n))Ñf(l) f(xφ(n))´f(x1φ(n))Ñ0 nÑ+8|f(xφ(n))´f(x1φ(n))| ěεą0 f [a,b] fn:RÝÑR xÞÝÑxn n fn RR n fn R+R+ n fn R x n R+ x rq? x pr=q? x p q? x p p q xp/q x 6 a (´2)2=6?

4ą03?

´2ă0

(´2)2/6(´2)1/3 xr xą0rPQ x,x1ą0r,r1PQ x rxr1=xr+r1x´r=1 x rx0= 1 (xx1)r=xrx1r r= 0xÞÑxr 1 xą0 αPR (rn)nPNPQN α (xrn)nPN

α (rn)nPN

xα=nÑ+8xrn αPQ

αă0

x,x1ą0α,βPR x

0= 1xαxβ=xα+β(xα)β=xαβx´α=1

x

α(xx1)α=xαx1α

αą0 xÑ0xα= 0 xÑ+8xα= +8

xÞÑax R aą1 a= 1 aă1 un= 1+1 2! +1 3! +¨¨¨+1 n! ePRzQ

2ăeă3

ex´1 x

ÝÝÝÑxÑ01 xÞÑex

R eą2ą1

R]0,+8]

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