[PDF] Chapitre8 : Fonctions continues



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Chapitre8 : Fonctions continues

I RǶ

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) f D f x0D Ƕ @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 fǶ [0,2] fǶ [0,1]Ƕ Ƕ 1 f |[0,1] f [0,1] f Af Bùñf AYB

Ƀ f [0,1] Ƭf|[0,1]

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

Ƕ f Ƕ x0PD

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| Ƕ 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

I Ƕ I R C0(I,R)

C0(I,R) C0I ĿC ŀ

f: [a,b]ÑRaăb d f(a)f(b)Ƕ f(a)ďdďf(b)f(b)ďdďf(a) cP[a,b] d=f(c)

I R f I f(I)

α,βPf(I) dαβ f(I)

αǶf(a) aPI

βǶf(b) bPI

dαβ aďb αβ aPIbPI [a,b]ĂII f I f [a,b] dǶf(c) ɍcP[a,b]ĂIǶ ĕ dPf(I) f(I) f(I) f(I) R f: [a,b]ÑR f([a,b]) [a,b] f([a,b]) f([a,b]) Ƕ

Ƕf(c) ɍcP[a,b]

a,bPR aăb f: [a,b]ÑR df(a)f(b) f(a)ďf(b) f(a)ďdďf(b) f(a)´dď0ďf(b)´d g: [a,b]ÝÑR xÞÝÑf(x)´d

Ĕ g [a,b] g(a)ď0ďg(b)

Ƕ Ƕ cP[a,b] g(c) = 0

f(a)ěf(b) g: [a,b]ÝÑR xÞÝÑd´f(x) ā (an)(bn) a

0=ab0=b

nPN g(an+bn 2 )ď0 an+1=an+bn 2 bn+1=bn an+1=anbn+1=an+bn 2 @nPN,aďanďbnďb (an) (bn) @nPN,bn´an=b´a 2 n @nPN,g(an)ď0g(bn)ě0 @nPN,g(an)ď0ďg(bn) (an)(bn) ā c aďcďb cP[a,b] g c g(c)ď0ďg(c) g(c) = 0 Ƕɍ Ƕ ĕ

ĕ fǶ

f(a) f(b)

ĕ Ƕ Ƕ cĘ Ƕ

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)

β Ƕ (yn)nPNǶ J

nPN xnP[a,b] yn=f(xn) (xn)nPN (x1n)nPN= (xφ(n))nPN l=nÑ+8xφ(n) lP[a,b]@nPN,aďxφ(n)ďb (yφ(n))nPN Ƕ β Ƕ f(l)@nPN,yφ(n)=f(xφ(n)) f

β=f(l) βPJ

āαPJ

J

Ƕ Ƕ R

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[

Ƕ Ƕ Ƕ f:xÑx

Ƕ Ƕ Ƕ f:xÑx[0,π]Y

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