Funções Reais de Variável Real

 

PBKS@_^KJFJ@ NFZÞEKNF J@ MIÌFMOKVP@ CFNPEJFJ@ J@ NKÇBNKF^ ^INKFK^ @ XIEÍZKNF^ Fjmkbkstrfìîi ` L`stîi Dispktfefr  Mft`mætknf :i Fbi ^`mk-Xr`s`bnkfe ^`mk-Xr`s`bnkfe

 CPBÌÐ@^ _@FK^ J@ SF_KÆS@E _@FE

Bim`s7 N`enki Oib` _`mkss`

Vu`ekmfb`, Mfki j` ?8?6

 

PBKS@_^KJFJ@ NFZÞEKNF J@ MIÌFMOKVP@ CFNPEJFJ@ J@ NKÇBNKF^ ^INKFK^ @ XIEÍZKNF^

.CPBÌÐ@^ _@FK^ J@ SF_KÆS@E _@FE

Zrfofedi Nk`btíckni j` Nfrænt`r Fvfekftkvi Fvfekftkvi f s` s`rr `b `btr tr`l `lu` u` Pb Pbkv kv`r `rsk skjf jfj` j` Nf Nftþ tþek eknf nf j` Miìf Mi ìfmo mokq kqu` u` bf Cf Cfnu nuej ejfj fj`` j` Nk Nkçb çbnk nkfs fs ^inkfk ^in kfkss ` Xiektk Xiektknf nfss bf Nfj`kr Nfj`krff Mft`m Mft`mætk ætknf nf e`nnkibfjf p`ef7

Vu`ekmfb`, Mfki j` ?8?6

?

 

Íbjkn` Kbtrijuìîi............................. Kbtrijuìîi....... ............................................ ............................................ ............................................ .......................................................... ....................................:: ?. CPBÌÐ@^ _@FK^ J@ SF_KÆS@E SF_KÆS@E _@FE.............. _@FE.................................... .............................................................0 .......................................0 ?.6. J`ckbkìð`s ` priprk`jfj`s `e`m`btfr`s `e`m`btfr`s sior` cubìð`s r`fks j` vfrkæv`e r`fe...................... r`fe.................. ....00 ?.?. @x`mpeis j` cubìð`s..... cubìð`s........................... ............................................ ............................................ ...................................................... ....................................4 ....4 ?.?.6. Cubìð`s Xiekbimkfks....... Xiekbimkfks............................. ............................................ ............................................ ...................................................... ................................44 ?.?.?. Cubìð`s _fnkibfks..... _fnkibfks........................... ............................................ ............................................ ............................................... ................................... ..........68 68 ?.?.>. Cubìîi @xpib`bnkfe ` Eilfrítmknf.......................................... Eilfrítmknf....................................................................... ...................................... .........66 66 ?.?.:. Cubìð`s Zrklibimëtrknfs ` Zrklibimëtrknfs Kbv`rsfs.................................................. Kbv`rsfs...................................................6: .6: Okoekilrfckf.................................. Okoekilrfckf............ ............................................ ............................................ ............................................ ............................................. ...........................63 ....63

>

 

Kbtrijuìîi I pr`s pr`s`btf `btf trfofedi trfofedi suoirjkbf suoirjkbf-s` -s` fi t`mf t`mf77  CPBÌÐ@^ _@FK^ J@ SF_KÆS@E _@FE. Jkz`r  qu` umf

cub cubìî ìîii ë umf fpekn fpeknfìî fìîii `btr` `btr` nib nibgub gubtis tis.. Fs cub cubìð ìð`s `s j`snr` j`snr`v`m v`m c`bþm`bi c`bþm`biss

bumërknis ` pij`m r`pr`s`btfr-s` ftrfvës j` lræcknis sior` `kxis nfrt`skfbis. I lræckni j` umf cubìîi p`rmkt` v`r, mukti cfnkem`bt`, tijf f suf `vieuìîi. Xirëm, pir v`z`s, pij` s`r  mfks nþmiji trfofedfr nim f `qufìîi iu cþrmuef jf cubìîi, gæ qu` nim `ef t`mis è bissf jksp jkspis iskì kìîi îi i nibg nibgub ubti ti j` ip`r ip`rfì fìð` ð`ss qu` qu` j` j`v` v`mi miss fp fpekn eknfr fr è vf vfrk rkæv æv`e `e kb kbj` j`p` p`bj bj`b `bt` t`,, birmfem`bt` r`pr`s`btfjf pir x, pfrf iot`r f vfrkæv`e j`p`bj`bt`, birmfem`bt` r`pr`s`btfjf  pir y. Xij`mis Xij`mis kmflkbfr qu` umf cubìîi ë umf mæqukbf `m qu` kbtrijuzkmis um b÷m`ri x ji nibgubti j` pfrtkjf, j`ef sfkbji i b÷m`ri c(x). Vufbti f m`tijieilkf jkz`r pfrf nibnr`tkzfìîi ji pr`s`bt` trfofedi r`nirr`u-s` f nibsuetf j` ekvris, frtklis qu` nibtkbdfm f kbcirmfìîi ji t`mf, p`squksf bf kbt`rb`t, tënbknf j` r`sumi ` jklktfìîi, fsskm nimi fs p`squksfs ` okoekit`nfs nim i kbtukti j` trfz`r i `ss`bnkfe ` m`edirfr i j`s`bvievkm`bti nk`btíckni ji trfofedi. @ss` mëtiji ë umf r`vksîi ji `stfji jf frt` ji t`mf, i qufe p`rmkt` kj`btkcknfr nimi ` ibj` `e`s `stîi jkspibív`ks, bi ïmokti fnfjëmkni-nk`btíckni. Xfrf irlfbkzfr f p`squksf, ousniu-s` puoeknfìð`s `m ekvris ` frtklis nk`btícknis qu` foirjfm fs t`mætknfs prkbnkpfks j` lëb`ri.

:

 

?. CPBÌÐ@^ _@FK^ J@ SF_KÆS@E _@FE Kbtrijuìîi fi `stuji j` cubìð`s cubìð`s r`fks j` vfrkæv`e r`fe7 biìð` biìð`ss oæsknfs ` `x`mpeis1 `x`mpeis1 cubìð`s trklibimëtrknfs kbv`rsfs. Nimi `st` ë um nibt`÷ji nim qu` tijis is feubis gæ tkv`rfm felum nibtfti prëvki, bîi s`rîi  pists ofs`kffpr`s`btfjis muktis `x`mpeis pfrf fs biìð`s mfks `e`m`btfr`s. Nfjf um j`st`s fpr`s`btfjis j`st`s pists  ofs`kfs` bi qu` fnibt`n`u fnibt`n`u jurfbt` umf fuef ji nursi, ibj` i t`mpi jkspi jkspibív` bív`ee iorklf iorklf f felumfs felumfs ipìð`s bf fpr`s`btfìîi.

?.6. J`ckbkìð`s ` priprk`jfj`s priprk`jfj`s `e`m`btfr`s sior` cubìð`s r`fks j` vfrkæv`e r`fe Jfjis jiks nibgubtis

` , jkz`mis qu`

f tijf f nirr`spibjçb nirr`spibjçbnkf nkf `btr` `e`m` e`m`bbti

j` .

`

qu` f nfjf `e`m`bti j`

_` _`ppr`s r`s`btf `btfmi miss

`snr `snr`v `v`m `mis is

ë umf cubìîi  j`ckbkjf nim vfeir`s j`

`st stff

nirr nirr``spi pibj bjççbn bnkf kf

pi pirr

`

tf tfmo moëëm

cf cfz` z`mi miss ni nirr` rr`sp spib ibj` j`rr i

.

Fi `e`m`bti ndfmfmis iog`ti ` fi `e`m`bti Fi nibgubti

cfz nirr`spibj`r um ` um sþ

pf pfrf rf kbjk kbjknf nfrr qu` qu` fi `e `e`m `m`b `bti ti

`e`m`bti

pfrf

ndfmfmis jimíbki j`

ndfmfmis kmfl`m  j` .

(`snr`v`mis

) ` fi nibgubti

ndfmfmis nibgubti j` nd`lfjf j` . Ndfmfmis nibtrfjimíbki j` kmfl`bs (`snr`v`mis

fi nibgubti jfs

), iu s`gf, fi nibgubti jis `e`m`btis qu` sîi kmfl`m p`ef

cubìîi jis `e`m`btis ji jimíbki, i qufe ë bfturfem`bt` suonibgubti j`

` pij` s`r

r`pr`s`btfji pir 

Jkz`mis qu`

ë umf cubìîi umf cubìîi r`fe j` vfrkæv`e r`fe  (crvr) qufbji

`

sîi suonibgubtis

j` . J` flirf `m jkfbt`, f m`bis qu` s`gf jkti feli `m nibtrærki, nibskj`rfr`mis s`mpr` cubìð`s r`fks j` vfrkæv`e r`fe, qu`, pfrf skmpekcknfr i t`xti, pfssfr`mis f trftfr skmpe`sm`bt` pir cubìð`s. Ndfmfmis lræckni jf cubìîi

fi nibgubti . 0

 

Jkz` Jkz`mi miss qu` i

ë umf cub cubìî ìîii ekmktfjf s` `x `xks kst` t`

. Xir iutrfs pfefvrfs,

jkz`mis qu`

tf tfee qu` qu`

ë umf cubìîi ekmktfjf s`

ë umf cubìîi mfgirfjf/mkbirfjf mfgirfjf/mkbirfjf  s`

, pfrf pfrf tiji tiji

ë um nibgubti ekmktfji. Zfmoëm i cir `bqufbti nibgubti. ndfmfmis ndfmfmis supr`mi/   supr`mi/ 

J` miji fbæeili, fi supr`mi/ íbckmi /mæxkmi/ míbkmi ji nibgubti íbckmi/ mæxkmi/ míbkmi  míbkmi j` , s`mpr` qu` `xkstkr`m. @x`mpei7

@x`mpei7 F cubì cubìîi îi

, ni nim m

t` t`m m ni nibt btrf rfji jimí míbk bkii

. Ë umf umf cubì cubìîi îi

mkbirfjf, mkbir fjf, t`m míbkm míbkmii , mfs bîi ë umf cubìîi mfgirf mfgirfjf, jf, p`ei qu` bîi ë, `btî `btîi, i, umf cubìîi ekmktfjf.

Jkz`mis qu` umf cubìîi  ë nr`sn`bt` nr`sn`bt`  s` s`mpr` qu`

tkv`rmis

1



ë j`nr`sn`bt` s` s`mpr` qu`



ë `strktfm`bt` nr`sn`bt` s` s`mpr` qu`



j`nr`sn`bt` s` s`mpr` qu` ë `strktfm`bt` j`nr`sn`bt` 



ë mibþtibf s` ë nr`sn`bt` iu j`nr`sn`bt`1



ë `strktfm`bt` mibþtibf s` mibþtibf s` ë `strktfm`bt` nr`sn`bt` iu `strktfm`bt` j`nr`sn`bt` j`nr`sn`bt`..

@x`mpei7 F

cubì cubìîîi

ë

`strktfm`bt` nr`sn`bt` bi kbt`rvfei

tkv`rmis

`st strk rktf tfm` m`bt bt``

1 tkv`rmis

1

tkv`rmis

j`n `nr` r`sn sn``bt bt``

1

bi

kb kbt` t`rv rvfe feii

`

. F suf r`pr`s`btfìîi r`pr`s`btfìîi lræcknf tirbf mfks kbtuktkvf

f j`snrkìîi jf mibitibkf jf cubìîi7

3

 

Jkz`mis qu` umf cubìîi   ë pfr s` 

pfrf tiji i

 ë ímpfr s`

1

pfrf tiji i

.

I lræckni j` umf cubìîi pfr ë skmëtrkni `m r`efìîi fi `kxi jfs irj`bfjfs, `bqufbti qu` i lræckni j` umf cubìîi ímpfr ë skmëtrkni `m r`efìîi è irkl`m.

@x`mpei7 F cubìîi

ë umf cubìîi pfr, piks

I s`u s`u lræck ræcknni ë sk skm mëtrkn trknii `m r`ef r`efììîi fi `kx kxii jf jfss irj rj``bfj fjffs. Xir iu iutr trii ef efjji, f cubìîi

ë umf cubìîi ímpfr, piks  „ 

Jkz`mis qu` umf cubìîi  pfrf tiji i

 cubìîi p`rkþjknf j` p`ríiji ë umf umf cubìîi

s`

.

I lræckni j` umf cubìîi p`rkþjknf j` p`ríiji

r`p`t`-s`  j` r`p`t`-s`

`m

ubkjfj`s ji `kxi jfs

fonkssfs, fssinkfji fis iog`tis jf cubìîi.

@x`mpei7 F cubìîi

ë p`rkþjknf j` p`ríiji

, piks pfrf qufequ`r

,

S`r`mis fjkfbt` mfks felumfs kbcirmfìð`s sior` `stf cubìîi. Jfjf

umf

cubìîi

,

fis

`e`m`btis

tfks

qu`

ndfmfmis z`ris  j` . Is z`ris sîi fs fonkssfs jis pibtis   j` kbt`rs`nìîi ji lræckni jf cubìîi nim i `kxi .@x`mpei7 J`t`rmkb`mis is z`ris jf cubìîi

7 .

I nibgubti jis z`ris j`

ë

.

Jkz`mis qu` umf cubìîi <

 



ë kbg`tkvf s` pfrf tiji i

tfks qu`

tkv`rmis



ë sior`g`tkvf s` pfrf tiji i

`xkst`

tfe qu`



ë okg`tkvf s` cir kbg`tkvf ` sior`g`tkvf.

1 1

Pmf cubìîi ë kbg`tkvf s` i s`u lræckni kbt`rs`tf qufequ`r r`tf dirkzibtfe, bi mæxkmi, umf v`z. @x`mpei7

@x`mpei7 F cubìîi

bîi ë kbg`tkvf piks , pir `x`mpei

. _`pfr`-

s` qu` nibs`lukmis cfnkem`bt` trfìfr umf r`tf dirkzibtfe qu` kbt`rs`t` i lræckni jf cubìîi `m jiks pibtis jkstkbtis. F cubìîi qu` r`suetf jf r`strkìîi j` ë kbg`tkvf. QJ`ckbkmis f r`strkìîi j` qu`

f

pfrf nfjf

^`gfm ^`gf m

, f qufe r`pr`s`btfmis pir

fi kbt`rvfei , è cubìîi j`

tfe

.] `

jufs cubìð`s cubìð`s tfks qu`

J`ckbk J`c kbkmis mis f cubìîi nimpistf j` nim , f cub cubìî ìîii j`sklb j`sklbfjf fjf pir ë

`m

` pfrf nfjf

,

.

, nugi nugi jimíbk jimíbkii

.

Jfjf umf Jfjf umf cubì cubìîi îi kbg` kbg`tk tkvf vf , j` j`ck ckbk bkmi miss f  cubìîi cubìîi kbv`r kbv`rsf sf j` , ni nimi mi s`bji tfe qu` pfrf tiji i 1 fsskm, . _`pr`s`btfr`mis _`pr`s`btfr`mis f cubìîi kbv`rsf j` pir . Xir `x`mpei, f cubìîi kbv`rsf j` ë . I lræckni j` r`suetf ji lræckni j` cfz`bji umf skm`trkf `m r`efìîi è r`tf . ?.?. @x`mpeis j` cubìð`s ?.?.6. Cubìð`s Xiekbimkfks Jkz`mis qu` umf cubìîi ë piekbimkfe ë piekbimkfe j` lrfu b s` b s` t`m f cirmf

4

 

ibj`

sîi nibstfbt`s r`fks, ndfmfjfs ni`cknk`bt`s.

@m pfrtknuefr, s` umf cubìîi piekbimkfe t`m lrfu z`ri, um, jiks iu trçs, ë ndfmfjf j` cubìîi j` cubìîi nibstfbt`, fckm, qufjrætknf iu n÷oknf iu n÷oknf,, r`s r`sp` p`tkv tkvfm` fm`bt` bt`.. S`gfm S`gfmis is is s`lukb s`lukbt`s t`s `x`mpeis7

Cubìîi nibstfbt` (

)

  I s`u lræck lræckni ni ë uumf mf r`t r`tff pfr pfrfe`ef fe`ef fi `kxi `kxi jfs fonks fonkssfs sfs ` qu` nnibtëm ibtëm i pib pibti ti (8, ).

Cubìîi fckm (

)

  I s`u lræck lræckni ni ë umf r`tf qu` kbt`rs`tf kbt`rs`tf i `kxi jfs irj`bfjf irj`bfjfss (`kxi j`nekv` .

Cubìîi qufjrætknf (

) bi pibt pibtii

` t`m

)

I s`u lræckni ë umf pfræoief j` `kxi v`rtknfe ` nibnfvkjfj` nibnfvkjfj` vietfjf pfrf nkmf s` nibnfvkjfj`` vietfjf pfrf ofkxi s` nibnfvkjfj

, iu

1 f pfræoief kbt`rs`tf i `kxi jfs fonkssfs bis z`ris

jf cubìîi.

;

 

?.?.?. Cubìð`s _fnkibfks

Jkz`mis qu` umf cubìîi r`fe j` vfrkæv`e r`fe ë rfnkibfe s` pij` s`r `snrktf bf cirmf

ibj`

`

sîi piekbþmkis `m . I jimíbki j` umf cubìîi rfnkibfe nim `stf cirmf ë

i nibgubti j` tijis is b÷m`ris r`fks, `xn`pti fqu`e`s pfrf is qufks s` t`m 

@x`mpei7

.

.

I s`u jimíbki ë . Pmf r`pr`s`btfìîi ji s`u lræckni ë

@x`mpei7   I s`u jimíbki ë

, nim

.

. ^`lu`m-s` r`pr`s`btfìð`s jis s`u lræcknis7

 

?.?.>. Cubìîi @xpib`bnkfe ` Eilfrítmknf

68

 

^`gf $eft`x fUkb{Umftdoo _}R+Us`tmkbusU{6U}. Ndfm Ndfmfmis fmis cubìî cubìîii `xpib` `xpib`bnkf bnkfee j` ofs` ofs` è cubìîi r`fe j` vfrkæv`e r`fe .

I s`u jimíbki ë ` i nibtrfjimíbki ë . F cubìîi ë kbg`tkvf ` suf mibitibkf j`p`bj` ji vfei vfeirr j` . @m pfrtk pfrtknu nuef efr, r, ss`` f ccub ubìî ìîii ë ``st strkt rktfm fm`b `bt` t` nr` nr`sn sn`b `bt` t`,, `bq `bquf ufbt btii qu` qu` s`

f cubìîi ë `strktfm`bt` j`nr`sn`bt`. j`nr`sn`bt`.

@x`mpei7 _`pr`s`btfìð`s jis lræcknis j` felumfs cubìð`s `xpib`bnkfks nim ofs` mfkir qu` 67

 Bîi nibcubjkr nim . Q@x`rnínki7 Nimpfrfr is lræcknis j` ` .] @x`mpei7 _`pr`s`btfìð`s _`pr`s`btfìð`s jis lræcknis j` felumfs cubìð`s `xpib`bnkfks nim ofs` m`bir qu` 67

@x`mpei7 _`pr`s`btfì _`pr`s`btfìð`s ð`s jis lræcknis jfs cubìð`s `xpib`bnkf `xpib`bnkfks ks j` ofs` `

7

66

 

_`nirj _`n irj`mi `miss fel felumf umfss pri priprk prk`jf `jfj`s j`s jfs pitçb pitçbnkf nkfss j` b÷m`ri b÷m`riss r`fks. r`fks. ^`gfm ^`gfm `

. 6.

@x`mpei7

?.

@x`mpei7

>.

@x`mpei7

:.

@x`mpei7

0.

@x`mpei7

3.

@x`mpei7

.

pfrf tiji i 6>

 

:. 0.

(mujfbìf j` ofs`) `

?.?.:. Cubìð`s Zrklibimëtrknfs ` Zrklibimëtrknfs Zrklibimëtrknfs Kbv`rsfs ^`bi ` frni s`bi Nibskj Nib skj`r` `r`mis mis f cub cubìî ìîii s`bi jf jfjf jf pi pirr nibtrf nib trfji jimíb míbki ki

,

. I s` s`uu ji jimí míbk bkii ë

. Zrf Zrftftf-s` s` j` umf cub cubìî ìîii ímpfr, ímpfr, gæ qu`

` t` t`m m .

Pmf r`pr`s`btfìîi ji s`u lræckni ë

F cubìîi s`bi „ L`iL`orf Ciedf Lræcknf Jkbïmknf

^`gf

` nibskj`r`m nibskj`r`mis is f r`strkìîi jf cubìîi s`bi f `st` kbt`rvfei,

j`sklbfmis pir r`strkìîi prkbnkpfe . F cubìîi

, f qufe

ë kbg`tkvf, p`ei qu` pij`mis timfr f suf

s`bi7 cubìîi kbv`rsf frni s`bi7 cubìîi `ssf qu` f nfjf cfz nirr`spibj`r i frni nugi s`bi ë , t`m pir jimíbki jimíbki nibtrfjimíbki

. Pmf r`pr`s`btfìîi r`pr`s`btfìîi ji s`u lræckni ë

Xfrf nfjf

t`mis

`

. 6:

 

F cubìîi frni s`bi „ L`iL`orf Ciedf Lræcknf Jkbïmknf

 Bit`-s` qu` f `sniedf `sniedf ji kbt`rvfei qufequ`r næenuei.

ë umf m`rf nibv`bìîi nibv`bìîi ` bîi r`suetf j` j`

Okoekilrfckf

dttps7//nfenuok.wirjpr`ss.nim/?86 pr`ss.nim/?86>/68/63/fuef->-kbtrijunfi >/68/63/fuef->-kbtrijunfi-fs-cubni`s-r`fks-j` -fs-cubni`s-r`fks-j`-vfrkfv`e-vfrkfv`edttps7//nfenuok.wirj r`fe/ fn`ssfji r`fe/  fn`ssfji fis fis 68 j` Mfki j` ?8?6 fs 66d7 ?0m X`r`krf , _issfbf _issfbf M.M. ^ijrë, Peyss`s.  Mft`mætknf @ss`bnkfe @bskbi Cubjfm`btfe, Cubjfm`btfe, Mëjki ` ^up`rkir bi Orfske 

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