Oggiesercitasione in Aula 3 ( Dip . Matematicy

delle 16:00 Ile 10:00

f ,g funzionitimitdte in Iinterruo

dlf ,g)=supµfH-gal distantXEI

µfgH.

Una distant sum insieme X e'

una fanzined : Xxx - [ oiax t.ci

Qcxy ) to, di,y)⇐o# X=y

dcxy) = dcyix) Faye.X.

dcxiy ) = de,⇒+dEiy ) Hxiyite Xmodule : R

,con la dist

. day )=Ix - yl

Alto esempio :

1RWconladistanzadCx.yj-11e-y1l-E2fxi-yIfkSuRNpossiamauchedefinimeaHredHy1-IEBXi-yilveoificaoeoheiunadistaraBsdWoH-kaykR2.ds@ykaoDos31xHyK1HdpHytfIZ.l

×iTiP]% p€⇐⇒

Sia

CFI) linsiemedelkfunzionicontinue.%ktIinRdt,g) = jhplfcxs - gcxy tuna dithuta suC#d (fig ) xo

d(f,f) - 0

d(f,gko ⇒ ftp.lfa.gg/=o ⇒ f* - gatx

dtigtdyg,f) ownf=8

Distaangolare ?

d (fig ) € d (f ,h)+d@,g)

1fH-gcxYsffcxthaItlhH-gKyVxeINNmfplfjtthHIdCh.g1dCf.hp-VxeI1fa-gakdLf.htd@n.gjPassandedtnpsux.siottienedHigkdf.h

ltd (hg) .

CHI) , conquests diorama,e- mnoSpaziometncocioiuninsane mum 't di una distant

.

Ciano altreditrnx possible.

su I ( poseuffmtyg ,dff,g) = f±µga-gold ×

" distant e "

ddf,g)=ff±lfH-gaPdx]K"

diorama E"

Torniamo a dcfigkxheplfcx) -galDiremo che fu , f E L#= insieme delle funzionilimbte

su I

Direme che fn converged f uniformemente in I,

per n oakm dtn,fl = o.

n oo

ciei : Feb F he t.c.VN > ne d(fn,f)seFeb Fnett . fnonc, xsffllfkx) -fake

" " 1 1 th >nefx€I ,|fnH-fake

CONHNNIFORMEFE> 07 net .

c. th > ne that Bnd - fakeCONV .

PUNTUALE

#xeIFn,× t.c.VN >ns×Bnd.fake

fnk) = Xtn × €0,1 )Converge pmbualmente a fcx ) ± I

ma non uniform emente

dtmfkyyf,,,4

- MY -

1-10

converge uniform !t in ( x,

s ) and >°

n a

dtn,ftp.yyf.fi- x% ) - t - d% - o

d (fig ) < e ffkiaAristide=g*

Quindi ,

"fn funif ? "

signifies on,

fissato una fascia di semiampiessae into meal grfiadi f ,

sen e' abbastansa grande told il grsfio di fn

gidcerd' nella fascia considerate

TEOREMAT fn continue in I, fn fwniformementern I

Athos f ei continue .

DimFXEI He >o 28>0 te

. Aye I icrifiante lyxk 8 si he

ffcxhfcpke1 fitfcpf Ifa -frat Ifncxt . fight Ifkyl-f$1

A N

d (fn ,f) dtfn,f)Fission in mode he il Neil 30 addends Siano < ÷

th-f$K§ + t.ci-fn$| < E se Ix - yk 8Usrame la contimuhioh guests fn : FJ t.CH-yl < or

lfncx) -f#k±

TEORTMI ( passaggbd limit sotto f )

fn , f Riemann - integrand in [a,b]fn f unifuin Gb]Amr

§fµd×¥F[ fad 't

Woildjmabafnx )dx= fabpnismafnxpdx

DIM

oetfafnxldx-§fadx/=/§Cfna - fc×Ddx/= { IFNH-fa1÷.

< d(fn,f) (b-a) = 0 D d(fn,f )

Ptt.

La cowu uniform marked laden bbihta' ?

fnokrinabik in I, fn fumfk in I €2 f deorsbik

NO.

Esempae fna= ME in R

fncx ) fcx )=fE= 1×1 pcmtualm .

dcfnftyyp IFE - ixll . ftp.pxFE- × ) -

F.9 'a= Exe -

s=¥604 "

refn f uniftema f non denisbik

.

fn f unifh fn , f denwbbi .

I fi f ' ? he

Esempio fax sennHD- , fcx ) = o

d(fn,f)=sgp Henney - Fof!a= eoknx )

fK÷eYn

www.fr-yn

I intend limiisto

Teortme {fn ] suciedi funziomi C' ( I )

, supp mono che

1) fi converge anifkad una fmrione of .

2) FXOEI te. {fn(xD converge

A her fn converge wife in I.

motte, dethef ie hmte delle fn

,

f i di lane d# ,e ftp.go the

Studier le cow. puntuak e uniform di fa= I

¥ ftp.fna-o.fmtrim

CI inT.dk#l=Ta+p.nnI*=fnfE.1=

n¥÷

fix ' .in#MIxIIIxIn*nexyst .¥yg :dan = ,÷ .fr Kaia

Monde'

Conn. Wwf .

in R.

Per 6 these motin,

non ci pro'

esser convergent a

Uniform in intervals che " siawicinan " all'

originC

.U

.

in [ x ,too ) on a >o

.

? si !

sup find = find an 0kid ¥ pen abb

. grade He ↳£e£Tfn deaesc in [ x ,too

nhjmo £3 thnx,

dx - 0 perhifnaso uniftenk , }]