Digitally Continuous Functions * Laurence Boxer % Abstract In

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XGUGBGZE
[ \]^_`abc^d`]
ef ghijkflkmno pqrsto u fivwif il xifvwfywvz wj {w|kf li} lyfxvwifj ~kvkkf nw{wvum €wxvy}kj ‚kv yj
xumm jyxƒ lyfxvwifj „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ jkk ‘kxvwif ’ li} vƒk nk“fwvwif” ev jkk•j mw–kmz vƒuv vƒwj
fivwif wmm ~k il lyfnu•kfvum w•€i}vufxk wf u |u}wkvz il u€€mwxuvwifj wf €uvvk}f •uvxƒwf{ ufn w•u{k
€}ixkjjwf{o €u}vwxymu}mz wf vƒk jvynz il •u€€wf{j ~kvkkf nw{wvum w•u{kj uj }kjimyvwifj u}k •inw“kn
‘yxƒ •inw“xuvwif il vƒk yjyum fivwif il xifvwfywvz wj yjklym wf jvynzwf{ €}i€k}vwkj il nw{wvum
w•u{kj ufn vƒkw} v}ufjli}•uvwifj —w{wvum w•u{kj u}k •unk y€ il muvvwxk €iwfvjo ufn k wjƒ vi
~k u~mk vi •kujy}k vƒkw} {ki•kv}wx €}i€k}vwkj yjwf{ u ˜yxmwnkuf i} ivƒk} ™š •kv}wx ›ik|k}o vƒk
vi€imi{z wfnyxkn if u nw{wvum w•u{k ~z jyxƒ u •kv}wx wj nwjx}kvkœ wf jyxƒ u vi€imi{zo
 k|k}z lyfxvwif ~kvkkf nw{wvum w•u{kj wj xifvwfyiyjœ
 u •užw•um xiffkxvkn jkv wj u jwf{mk €iwfvœ
 vƒk xifvwfyiyj w•u{k il u nw{wvum mwfk jk{•kfv ™ •uz ~k ufz jy~jkv Ÿ il vƒk }uf{k j€uxk jyxƒ
vƒuv Ÿ ¡ ™ œ
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kvx ƒyjo ƒwmk k wjƒ vi yjk jvufnu}n •kv}wxj vi nkvk}•wfk {ki•kv}wx lkuvy}kj il nw{wvum w•u{kjo
k ni fiv yjk vƒk vi€imi{z wfnyxkn ~z jyxƒ u •kv}wx vi nkvk}•wfk €}i€k}vwkj jyxƒ uj xiffkxvknfkjj
wf u nw{wvum w•u{kœ jw•wmu}mzo k ni fiv yjk vƒk vi€imi{z wfnyxkn ~z u •kv}wx vi nkvk}•wfk vƒk xmujj
il lyfxvwifj ~kvkkf nw{wvum w•u{kj vƒuv k xifjwnk} vi ƒu|k u€€}i€}wuvk xifvwfywvz mw–k €}i€k}vwkj
ef vƒk xy}}kfv €u€k}o k kžvkfn vƒk }kjymvj il ghijkflkmno pqrst ~z kžu•wfwf{ nw{wvum |k}jwifj
il jk|k}um xmujjwxum fivwifj l}i• vi€imi{zo wfxmynwf{ ƒi•ki•i}€ƒwj•o }kv}uxvwifo kžvkfjwif il u
xifvwfyiyj lyfxvwifo ufn ƒi•ivi€z
_ d d] _d ‚kv nkfivk vƒk jkv il wfvk{k}j k nkfivk ~z vƒk jkv il muvvwxk €iwfvj wf u ~iyfnkn xy~k wf
˜yxmwnkuf nw•kfjwifum j€uxk ‰ˆ‡‡…‹ Œ…‡ wj u €iwfv kuxƒ il ƒijk xii}nwfuvkj wj uf wfvk{k}
i} jw•€mwxwvzo k limmi ghijkflkmno pqrst ~z ujjy•wf{ vƒk}k wj u €ijwvw|k wfvk{k} jyxƒ vƒuv
 ” ¡ li} p ev wj yjklym vi nwjvwf{ywjƒ ~kvkkf u „…†…‡ˆ‰ …‹‡Ž ufn u „…†…‡ˆ‰ …! ˆ†  immiwf{ ghijkflkmno pqrsto
k juz wj u „…†…‡ˆ‰ …‹‡Ž  „…†…‡ˆ‰ …! ˆ† wj u jy~jkv il 
i €iwfvj " " " ” ufn # # # ” il u nw{wvum w•u{k u}k  …† $ %Œ  wl ufn ifmz wl
li} ji•k xii}nwfuvk & o " ' #' p ufn li} umm xii}nwfuvkj ( & o " #  j wf ghijkflkmno pqrsto
k ujjy•k ufz •kv}wx ) nk“fkn if vƒuv k yjk wmm juvwjlz
 ) " # ” * p wl " ufn # u}k nwjvwfxv €iwfvj il o ufn
 ) " # ” p wl ufn ifmz wl " ufn # u}k fkw{ƒ~i}j
i} kžu•€mko ~ivƒ vƒk ˜yxmwnkuf ufn xwvz ~mix– •kv}wxj ƒu|k vƒkjk €}i€k}vwkj ƒkf vƒk}k wj fi
fkkn vi j€kxwlz nw•kfjwifo k }wvk ) li} ) 
+ ,- ./0/1. 234 ghijkflkmno pqrst 5 ‡ 6 7 ˆ„ 8 7 9 % „…†…‡ˆ‰ … ! ˆ†  : 5 ‡ ; < 6 = 8 % ˆ
> Ž‹‡…Œ : ?$  ; … nw{wvummz xifvwfyiyj uv @ 6 …> ˆ„ Œ‰Š …> > Œ A Š B * p ‡$ … ˆ C * p
Ž‹$ ‡$ ˆ‡
6 ˆ„ ) @ ” ¡ C … ! ‰…  )9 ; @ ” ;  ”” ¡ B D ˆŠ ; … nw{wvummz xifvwfyiyj …> ˆ„ Œ‰Š …> ; … „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ ˆ‡ A Š 6 :
’
nw{wvum w•u{k 6 wj „…†…‡ˆ‰‰Š ‹Œ ‹‡ „ wl ufn ifmz wl li} k|k}z €uw} il €iwfvj 7 6 o ( o
vƒk}k kžwjvj u jkv @ 7 6 jyxƒ vƒuv @ o o ufn ufn u}k fkw{ƒ~i}jo
p p 
‚kv 7 o  ƒk jkv
g t ¡ ¡ wmm ~k xummkn u „…†…‡ˆ‰ …‡ A ˆ‰
ghijkflkmno pqrst nk•ifjv}uvkj u fy•~k} il lyfnu•kfvum €}i€k}vwkj il nw{wvummz xifvwfyiyj lyfx
vwifjo wfxmynwf{ vƒk limmiwf{
p , 1 .,/ 1 el ; < 6 = 8 wj nw{wvummz xifvwfyiyj ufn " ufn # u}k fkw{ƒ~i}j wf 6 o
vƒkf kwvƒk} ; " ” ; # ” i} ; " ” ufn ; # ” u}k fkw{ƒ~i}j wf 8 
’ fz xifjvufv lyfxvwif ~kvkkf nw{wvum w•u{kj wj nw{wvummz xifvwfyiyj
 jy• il nw{wvummz xifvwfyiyj lyfxvwifj fkkn fiv ~k nw{wvummz xifvwfyiyj
 lyfxvwif ; < 6 = 8 ~kvkkf nw{wvum w•u{kj wj nw{wvummz xifvwfyiyj wl ufn ifmz wl vƒk w•u{k
yfnk} ; il k|k}z nw{wvummz xiffkxvkn jy~jkv il 6 wj nw{wvummz xiffkxvkn
 nw{wvummz xifvwfyiyj lyfxvwif ~kvkkf nw{wvum wfvk}|umj ƒuj vƒk ‡ ! „…ˆ‡ ˆ‰Ž Œ ‡Š
nnwvwifummzo k ƒu|k vƒk limmiwf{
1 1 /0/1. 232 ?$ ‹Œ ! Œ…‡…Œ Œ> „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ … ‹Œ‡…ŽŒŽ wk …> ; < 6 = 8
ˆ„ < 8 = ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ …‡$ 6 7 8 7 9 ˆ„ 7 š ‡$  ‡$ ‹Œ ! Œ…‡ > Ž‹‡…Œ ; < 6 = … „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ :
ŒŒ> k •w•wx vƒk €}iil vƒuv vƒk xi•€ijwvwif il xifvwfyiyj lyfxvwifj wj xifvwfyiyj ‚kv
@ 6 ufn mkv @ ; @ ” ‚kv B * p ~k {w|kf ‘wfxk wj xifvwfyiyj uv @ o vƒk}k kžwjvj C * p jyxƒ
vƒuv wl 8 ufn )9  @ ” ¡ C o vƒkf )š   @ ”  ”” ¡ B  ‘wfxk ; wj xifvwfyiyj uv @ o vƒk}k kžwjvj
C * p jyxƒ vƒuv wl 6 ufn ) @ ” ¡ C o vƒkf )9  @ ;  ”” ¡ C  ev limmij vƒuv wl 6 ufn
) @ ” ¡ C o vƒkf ) š  ; @ ”” ;  ””” ¡ B 
‘wfxk B uj u}~wv}u}zo ; wj nw{wvummz xifvwfyiyj uv @  ‘wfxk @ uj uf u}~wv}u}z €iwfv il 6 o
; wj nw{wvummz xifvwfyiyj !
d d^ ` ` `_ d ef vi€imi{zo u lyfxvwif ; < 6 = 8 vƒuv wj xifvwfyiyjo ifkvi ifko ufn ifvio jyxƒ vƒuv vƒk wf|k}jk
lyfxvwif ; < 8 = 6 wj xifvwfyiyjo wj xummkn u $ Œ ! Œ! Œ $ … ! o wf ƒwxƒ xujk 6 ufn 8 u}k xummkn
$ Œ ! Œ ! Œ $ …‹ ›i•ki•i}€ƒwx j€uxkj u}k }k{u}nkn uj vi€imi{wxum xi€wkj il kuxƒ ivƒk} ƒwj jy{{kjvj
vƒk limmiwf{
+ ,- ./0/1. 34 5 ‡ 6 ˆ„ 8 % „…†…‡ˆ‰ … ! ˆ†  : 5 ‡ ; < 6 = 8 % ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ
‡$ ˆ‡ … Œ ‡Œ Œ ˆ„ Œ‡Œ : > ; < 8 = 6 … ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ ‡$  ; … ‹ˆ‰‰„ ˆ
nw{wvum ƒi•ki•i}€ƒwj• ˆ„ ˆŠ 6 ˆ„ 8 ˆ nw{wvummz ƒi•ki•i}€ƒwx :
1 1 /0/1. 32 …†…‡ˆ‰ $ Œ ! Œ ! Œ $ … ! … ˆ Ž… A ˆ‰ ‹ ‰ˆ‡…Œ ˆ ! Œ† „…†…‡ˆ‰ … ! ˆ†  wk …‡ …
… A Š ! ! ‡ …‹ ˆ„ ‡ ˆ…‡… A:
ŒŒ> ev wj kujwmz jkkf vƒuv li} k|k}z nw{wvum w•u{k 6 o vƒk wnkfvwvz lyfxvwif ) < 6 = 6 wj u
nw{wvum ƒi•ki•i}€ƒwj• ›kfxk nw{wvum ƒi•ki•i}€ƒwj• wj }k kžw|k
‚kv ; < 6 = 8 ~k u nw{wvum ƒi•ki•i}€ƒwj• ev wj kujwmz jkkf vƒuv ; < 8 = 6 wj u nw{wvum
ƒi•ki•i}€ƒwj• ›kfxk nw{wvum ƒi•ki•i}€ƒwj• wj jz••kv}wx
‚kv ; < 6 = 8 ufn < 8 = ~k nw{wvum ƒi•ki•i}€ƒwj•j ev limmij l}i• }i€ijwvwif ’’ vƒuv
; < 6 = wj u nw{wvum ƒi•ki•i}€ƒwj• ›kfxk nw{wvum ƒi•ki•i}€ƒwj• wj v}ufjwvw|k !
ifjwnk} vƒk limmiwf{
, 3 5 ‡ 6 ’ 7 ˆ„ ‰ ‡ 8 p 7 : 5 ‡ ; < 6 = 8 % ‡$ > Ž‹‡…Œ „  „
% Š ;  ” ’ : ?$  ; … ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ % … ‹‡…Œ % އ ; … Œ‡ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ : ?$ Ž
; … Œ‡ ˆ „…†…‡ˆ‰ $ Œ ! Œ ! Œ $ … ! : „ „ 6 ˆ„ 8 ˆ Œ‡ „…†…‡ˆ‰‰Š $ Œ ! Œ ! Œ $ …‹ ˆ 8 … „…†…‡ˆ‰‰Š
‹Œ ‹‡ „ ˆ„ 6 … Œ‡ :
ŒŒ> ˜ujzo ufn mklv vi vƒk }kunk} !
^_ c^d`] ]a ^ ] d`] ƒk ƒki}z il hkv}uxvj wj u •uvy}k u}ku il vi€imi{zo wf ƒwxƒ ƒyfn}knj il €u€k}j ƒu|k ~kkf €y~ mwjƒkn xmujjwx }klk}kfxk wj g i}jy–o pqs t ƒk fivwifj il ‡ ˆ‹‡…Œ ufn ‡ …Œ u}k lyfnu•kfvum
vi vƒk “kmn
lyfxvwif < 6 = 8 wj u ‡ ˆ‹‡…Œo ufn 8 wj u ‡ ˆ‹‡ Œ> 6 o wl
 8 7 6 o ufn
  ” li} umm 8 
ef vi€imi{zo }kv}uxvwifj {kfk}ummz u}k umji }k yw}kn vi ~k xifvwfyiyj k juz u lyfxvwif wj u „…†…‡ˆ‰
‡ ˆ‹‡…Œ wl wv wj u }kv}uxvwif vƒuv wj nw{wvummz xifvwfyiyjo wf ƒwxƒ xujk wvj w•u{k wj u „…†…‡ˆ‰ ‡ ˆ‹‡
il wvj ni•uwf
1 1 /0/1. 34 D $ ˆ A ‡ $ > Œ‰‰Œ …† :
: … A …‡Š A Š „…†…‡ˆ‰ … ! ˆ† … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> …‡ ‰> :
: ? ˆ…‡… A …‡Š 5 ‡ 6 7 8 7 $ … ˆ „…†…‡ˆ‰ … ! ˆ† : > 6 … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> 8 ˆ„
8 … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> ‡$  6 … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> :
ŒŒ>
ƒk “}jv ujjk}vwif limmij l}i• vƒk luxv vƒuv vƒk wnkfvwvz lyfxvwif if u nw{wvum w•u{k wj u
nw{wvum }kv}uxvwif ƒk jkxifn ujjk}vwif limmij l}i• }i€ijwvwif ’’ !
lyfxvwif ; < 6 = 8 ‡ „ i} … ˆ ‡ …Œ Œ> vƒk lyfxvwif ;@ < 6 @ = 8 wl
 6 @ 7 6 o ufn
 ;  ” ;@  ” li} umm 6 @ 
k ƒu|k vƒk limmiwf{
,1 , 32 5 ‡ 6 % ˆ „…†…‡ˆ‰ … ! ˆ† ˆ„ ‰ ‡ 6 @ 7 6 : ?$  6 @ … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> 6 …>
ˆ„ Œ‰Š …> A Š „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ ;@ < 6 @ = 8 $ ˆ ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ ‡ …Œ
; < 6 = 8 > Œ A Š „…†…‡ˆ‰ … ! ˆ† 8 :
ŒŒ>  y} u}{y•kfv •inw“kj vƒk €}iil il u jw•wmu} vƒki}k• li} xifvwfyiyj lyfxvwifj ‘kk g i}jy–o
pqs t”
‘y€€ijk vƒk}k wj u nw{wvum }kv}uxvwif < 6 = 6 @  ‚kv ;@ < 6 @ = 8 ~k u nw{wvummz xifvwfyiyj
lyfxvwif ƒkf ~z }i€ijwvwif ’’o vƒk xi•€ijwvwif ;@ < 6 = 8 wj u nw{wvummz xifvwfyiyj kžvkfjwif
il ;@ 
if|k}jkmzo jy€€ijk k|k}z nw{wvummz xifvwfyiyj lyfxvwif ;@ < 6 @ = 8 nk“fkn if 6 @ ƒuj u
nw{wvummz xifvwfyiyj kžvkfjwif ; < 6 = 8  ƒkfo wf €u}vwxymu}o vƒk nw{wvummz xifvwfyiyj wnkfvwvz
lyfxvwif ) < 6 @ = 6 @ ƒuj u nw{wvummz xifvwfyiyj kžvkfjwif < 6 = 6 @  ƒyj 6 @ wj u nw{wvum }kv}uxv
il 6  !
k ƒu|k vƒk limmiwf{
,1 , 3 Ž Œ 6 @ … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> 6 ˆ„ ; < 6 = 8 … ˆ „…†…‡ˆ‰ $ Œ ! Œ ! Œ $ … ! :
?$  ; 6 @ ” … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> 8 :
ŒŒ> ‚kv < 6 = 6 @ ~k u nw{wvum }kv}uxvwif
ƒkf ; ; < 8 = ; 6 @ ” wj u nw{wvum
}kv}uxvwifo ~z }i€ijwvwif ’’ !
‚kv ) wfnwxuvk vƒk %ŒŽ„ˆ Š il <
)  ” p jyxƒ vƒuv p ƒk limmiwf{ wj uf ufumi{ il vƒk vƒki}k• vƒuv u j€ƒk}k Ÿ vƒuv ~iyfnj u nw•kfjwifum ~umm wf ˜yxmwnkuf nw•kfjwifum j€uxk wj fiv u }kv}uxv il g—y{yfn wo pqsst
,1 , 3 Ž ! ’ : ?$  ) … Œ‡ ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> :
ŒŒ> €}iil il vƒk ufumi{iyj }kjymv wf vi€imi{z •u–kj €}kxwjk vƒk wfvywvw|k fivwif vƒuv u
xifvwfyiyj lyfxvwif xuffiv }w€ u ƒimk wf wvj ni•uwf y} nw{wvum |k}jwif il vƒwj }kjymv wj ~ujkn if
u jw•wmu} wfvywvwifo ƒwxƒ •uz ~k •unk €}kxwjk ~z uf u€€kum vi u xu}nwfumwvz u}{y•kfv
s
‘y€€ijk vƒk}k k}k u nw{wvum }kv}uxvwif < = )  ifjwnk} vƒk €iwfv " p p p p”
wf œ " wj u fkw{ƒ~i} il ~ivƒ " @  p p p” ufn " p p p” ‘wfxk " @ ” " @ ufn
" ” " o wv limmij l}i• vƒk nw{wvum xifvwfywvz il vƒuv
" ”  p p” ‘w•wmu}mzo vƒk €iwfv #  ’ p p p” juvwj“kj
# ”  p p p” io mkv ™ ~k vƒk nw{wvum mwfk jk{•kfv wvƒ kfn€iwfvj " ufn #  ƒkf ™” •yjv ~k u nw{wvummz
xiffkxvkn jkvo wvƒ ™” ¡ ™ ’ yv vƒwj wj w•€ijjw~mko jwfxk ufz nw{wvummz xiffkxvkn jkv
xifvuwfwf{ ~ivƒ " ” ufn # ” •yjv ƒu|k xu}nwfumwvz uv mkujv  ƒk xifv}unwxvwif w•€mwkj ) wj
fiv u nw{wvum }kv}uxv il  !
d d^ ` `^` ‘y€€ijk ; < 6 = 8 u}k xifvwfyiyj lyfxvwifj ~kvkkf vi€imi{wxum j€uxkj jyxƒ vƒuv vƒk}k wj u
xifvwfyiyj lyfxvwif < 6 g pt jyxƒ vƒuv li} umm 6 o  ” ;  ” ufn  p”  ” ƒkf
; ufn u}k $ Œ ! Œ‡Œ …‹o ufn wj u $ Œ ! Œ‡Œ Š % ‡  ; ufn  efvywvw|kmzo u ƒi•ivi€z ~kvkkf
xifvwfyiyj lyfxvwifj wj u xifvwfyiyj nkli}•uvwif il ifk wfvi ufivƒk}o ƒk}k wv wj ilvkf yjklym vi
wfvk}€}kv vƒk wfvk}|um g pt uj u vw•k €k}win ev wj nkjw}u~mk vi ƒu|k u nw{wvum |k}jwif il vƒwj fivwifo
uj wv jkk•j vi ~k lyfnu•kfvum li} jyxƒ €y}€ijkj uj ufw•uvwif wf xi•€yvk} {}u€ƒwxj
+ ,- ./0/1. 34 5 ‡ ; < 6 = 8 % „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ % ‡  „…†…‡ˆ‰ … ! ˆ†  ˆ„
Ž Œ ‡$ … ˆ Œ…‡…A …‡ † ˆ„ ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ
< 6 g t = 8
Ž‹$ ‡$ ˆ‡ > Œ ˆ‰‰ 6  ” ;  ” ˆ„  ”  ” : D ˆŠ … ˆ nw{wvum ƒi•ivi€z
~kvkkf ; ˆ„ ˆ„ ; ˆ„ ˆ nw{wvummz ƒi•ivi€wx wf 8 :
k jƒi wf vƒwj jkxvwif vƒuv nw{wvum ƒi•ivi€z ƒuj •ufz €}i€k}vwkj ufumi{iyj vi vƒijk il ƒi•i vi€z
…†…‡ˆ‰ $ Œ! Œ‡Œ Š … ˆ Ž…A ˆ‰‹ ‰ˆ‡…Œ ˆ! Œ† „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ :
ŒŒ> … A …‡Š i} k|k}z nw{wvummz xifvwfyiyj lyfxvwif ; < 6 = 8 ufn ufz €ijwvw|k wfvk{k}
o vƒk lyfxvwif < 6 g t = 8 nk“fkn ~z
 ” ;  ” li} umm  ” 6 g t wj kujwmz jkkf vi ~k u nw{wvum ƒi•ivi€z l}i• ; vi ; 
Š ! ! ‡ Š ‘y€€ijk li} ji•k €ijwvw|k wfvk{k} o vƒk lyfxvwif < 6 g t = 8 wj u nw{wvum
ƒi•ivi€z ~kvkkf nw{wvummz xifvwfyiyj lyfxvwifj ; < 6 = 8 ufn < 6 = 8  ƒkf vƒk lyfxvwif
< 6 g t = 8 nk“fkn ~z
 ”  ” li} umm  ” 6 g t wj u nw{wvum ƒi•ivi€z ~kvkkf ufn ; 
? ˆ…‡… A …‡Š ‘y€€ijk < 6 g @ t = 8 wj u nw{wvum ƒi•ivi€z ~kvkkf ; < 6 = 8 ufn
< 6 = 8 o ufn
< 6 g t = 8 wj u nw{wvum ƒi•ivi€z ~kvkkf ufn < 6 = 8  ƒkf vƒk
lyfxvwif < 6 g @ t = 8 nk“fkn ~z
 ”  ” @ ” wlwl gg @ @ t @ œ t wj u nw{wvum ƒi•ivi€z ~kvkkf ; ufn  !
ef vƒk limmiwf{o k nkfivk ~z 9 vƒk €iwfv wf 9 umm il ƒijk xii}nwfuvkj u}k  k umji fivk
vƒuv
9 9 g pt , 3 5 ‡ " " : 5 ‡ < 9 = š 9 š % ‡$ „…†…‡ˆ‰ ‡ ˆ‹‡…Œ „  „
%Š
 9 ”  š ” 5 ‡ < 9 = 9 % „  „ % Š
 9 ”  š  š 9 ”  9 ”” 1 1 /0/1.
32 r
$  ” •už ?$  … ˆ „…†…‡ˆ‰ $ Œ ! Œ‡Œ Š % ‡  ‡$ …„ ‡…‡Š ! ˆ ) < 9 = 9 ˆ„ ‡$ ! ˆ $ < š 9 š = 9 … ‡$ …‹‰Ž…Œ > Ž‹‡…Œ :
ŒŒ> ‘w•€mk ufn mklv vi vƒk }kunk} !
k juz u nw{wvum w•u{k 6 wj „…†…‡ˆ‰‰Š ‹Œ‡ ˆ‹‡… % ‰ wl wvj wnkfvwvz •u€ wj nw{wvummz ƒi•ivi€wx vi u
xifjvufv lyfxvwif wf 6 
, 3 Œ ˆ‹$ …‡ † 9 … „…†…‡ˆ‰‰Š ‹Œ‡ ˆ‹‡… % ‰ :
ŒŒ> ev wj kujz vi jkk vƒuv vƒk lyfxvwif < 9 = 9 nk“fkn ~z
 9 ”   ” 9 ”” ƒk}k vƒk lyfxvwif wj uj wf ˜žu•€mk  o nk•ifjv}uvkj vƒk nw{wvum xifv}uxvw~wmwvz il 9  !
1 1 /0/1.
3 > 6
… ˆ „…†…‡ˆ‰ …! ˆ† ‡$ ˆ‡ … Œ‡ „…†…‡ˆ‰‰Š ‹Œ ‹‡ „ ‡$  6 … Œ‡ „…†…‡ˆ‰‰Š
‹Œ‡ ˆ‹‡… %‰:
ŒŒ> ‘y€€ijk vƒk}k k}k u nw{wvum ƒi•ivi€z < 6 g " t = 6 jyxƒ vƒuv li} umm 6 ufn
li} ji•k @ 6 o  ” ufn  " ” @  ‚kv 6 ~k u €iwfv jyxƒ vƒuv @ ufn u}k fiv wf
vƒk ju•k nw{wvum xi•€ifkfvj il 6  ƒkf vƒk}k wj u gp " t jyxƒ vƒuv  p” ufn  ” u}k wf
nw k}kfv xi•€ifkfvj il 6  ›kfxko  p” ufn  ” u}k fiv fkw{ƒ~i}j ƒwj wj u xifv}unwxvwifo
jwfxk wj nw{wvummz xifvwfyiyj ƒk ujjk}vwif limmij !
,1 , 3 > 6
… „…†…‡ˆ‰‰Š ‹Œ‡ ˆ‹‡… %‰ ˆ„ 8 … ˆ „…†…‡ˆ‰ ‡ ˆ‹‡ Œ> 6 ‡$  8 … „…†…‡ˆ‰‰Š
‹Œ‡ ˆ‹‡… %‰:
ŒŒ> ‘wfxk 6 wj nw{wvummz xifv}uxvw~mko vƒk}k wj u nw{wvummz xifvwfyiyj lyfxvwif < 6 g " t =
6 jyxƒ vƒuv li} ji•k @ 6 ufn li} umm 6 o  ” ufn  " ” @  ‚kv < 6 = 8 ~k
u nw{wvum }kv}uxvwif ƒkf vƒk lyfxvwif < 8 g " t = 8 nk“fkn ~z  ”   ”” li} umm
 ” 8 g " t juvwj“kj vƒk limmiwf{
q
 wj nw{wvummz xifvwfyiyjo ~z }i€ijwvwif ’’œ
  ” li} umm 8 œ ufn
  " ” @ ” li} umm 8 
ƒyjo 8 wj nw{wvummz xifv}uxvw~mk !
‡ ˆ„ 8 % „…†…‡ˆ‰ …! ˆ†  ˆ„ Ž Œ < 6 g " t = 8 … ˆ „…†…‡ˆ‰ $ Œ ! Œ‡Œ Š :
 Œ ˆ‹ g t ‡$ …„Ž‹„ > Ž‹‡…Œ < 6 = 8 „  „ %Š
 ”  ” > Œ ˆ‰‰ 6
… „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ :
ŒŒ>
ƒwj limmij kujwmz l}i• —k“fwvwif ’p !
, 3 5 6
?$ > $ "
‡
g t = ) % ˆ „…†…‡ˆ‰ $ Œ! Œ‡Œ Š %‡  ‡$ …„ ‡…‡Š > Ž‹‡…Œ
Œ
ˆ„ ˆŒ‡ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ : ?$  > Œ A Š g " t ‡$ …„Ž‹„ > Ž‹‡…Œ
… ˆ „…†…‡ˆ‰ Œ Œ Œ …
ŒŒ> i} po vƒk ujjk}vwif limmij l}i• u jw•€mk •inw“xuvwif il vƒk €}iil il ˜žu•€mk  
i jy€€ijk p k €}ixkkn ~z wfnyxvwif if  i} o vƒk ujjk}vwif ƒimnjo jwfxk @ wj
vƒk wnkfvwvz lyfxvwif if ) o ƒwxƒ wj u nw{wvum ƒi•ki•i}€ƒwj•
‘y€€ijk vƒk}k kžwjvj & gp " t jyxƒ vƒuv li} ¡ & o wj u nw{wvum ƒi•ki•i}€ƒwj• k
•yjv jƒi vƒuv ' wj u nw{wvum ƒi•ki•i}€ƒwj•
z ‚k••u  o ' wj nw{wvummz xifvwfyiyj
kžvo k jƒi ' wj ifvi
‚kv @  ” )  k •yjv jƒi @ wj wf vƒk w•u{k il '  ‚kv ) ~k
jyxƒ vƒuv '  ” @  k u}k nifk wvƒ vƒwj xmuw• wl '  ” @ 
‘y€€ijk '  ”  & ” ( @  ƒk nw{wvum xifvwfywvz il w•€mwkj '  ” wj u fkw{ƒ~i} il @ 
wvƒiyv mijj il {kfk}umwvzo
'  ”  ” ƒk}k p p
, 3 5 < ) "
) $
$ ! ! $ !:
w}jv k xifjwnk} vƒk xujk
p
p
‚kv ) ~k jyxƒ vƒuv '  ”  p ” z vƒk wfnyxvw|k ƒz€ivƒkjwjo ' wj u
nw{wvum ƒi•ki•i}€ƒwj•o ji wj u fkw{ƒ~i} il  ‘wfxk ' wj nw{wvummz xifvwfyiyjo '  ” wj u fkw{ƒ~i}
ilo i} k yum vio '  ”  p ” mjio '  ”  & ” wj ~z vƒk nw{wvum xifvwfywvz il
” u fkw{ƒ~i} ilo i} k yum vio '  ” ƒkjk vi }kjv}wxvwifj if '  ” w•€mz '  ” @ 
ƒk ivƒk} xujkj u}k uj limmij
 po p ƒwj wj jw•wmu} vi vƒk “}jv xujk
 o p j u~i|ko jwfxk ' wj u nw{wvum ƒi•ki•i}€ƒwj• ufn wj nw{wvummz xifvwfyiyjo
vƒk}k wj u fkw{ƒ~i} il wf ) jyxƒ vƒuv '  ” @ 
 po ’ ƒwj wj jw•wmu} vi vƒk €}k|wiyj xujk
ev limmij vƒuv ' wj ifvi
‘wfxk ) wj u “fwvk jkvo vƒk luxv vƒuv ' wj ifvi w•€mwkj ' wj ifk vi ifk
‚kv ufn ~k fkw{ƒ~i}j wf )  y} u}{y•kfv vƒuv ' wj ifvi wj kujwmz •inw“kn vi jƒi vƒk}k
kžwjv ) jyxƒ vƒuv ufn u}k fkw{ƒ~i}jo '  ” o ufn '  ”  ev limmij
l}i• hijkflkmn j , 1 .,/ 1 €}i€k}vz vƒuv ' wj nw{wvummz xifvwfyiyj
›kfxko ' wj u nw{wvum ƒi•ki•i}€ƒwj• !
immiwf{ wj uf ufumi{ il }iyk} j vƒki}k• vƒuv j€ƒk}kj u}k fiv xifv}uxvw~mk g—y{yfn wo pqsst
1 1 3 Ž ! ’ : ?$  > Œ ˆ‰‰ …‡ †  ) … Œ‡ „…†…‡ˆ‰‰Š ‹Œ‡ ˆ‹‡… % ‰ :
ŒŒ>
ƒwj wj uf w••knwuvk xifjk ykfxk il ‚k••u r !
‡ ’ Œ ( ’ …> ; … ˆ „…†…‡ˆ‰‰Š ‹Œ‡…ŽŒŽ > Ž‹‡…Œ ‡$ ˆ‡ … „…†…‡ˆ‰‰Š
Œ Œ‡Œ …‹ ‡Œ ‡ …„ ‡…‡Š Ž‹‡…Œ ) < ) = ) ‡$  ; ) :
ŒŒ> ‘y€€ijk ; ( )  ƒk €}iil il ‚k••u r •uz ~k •inw“kn vi jƒi vƒuv wl ƒuj u €iwfv
vƒuv wjf v “žkno vƒkf fi €iwfv il ) wj “žkn ~z  ƒyjo vƒk}k kžwjv €ijwvw|k wfvk{k}j @ ¡ " ufn
u nw{wvummz xifvwfyiyj lyfxvwif < ) g " t = ) jyxƒ vƒuv
pp
1 1 34 5 :
>
$ ! $   ” li} umm ) œ
  ” li} ¡ @ œ
  @ ” ( 
‘wfxk wj u nw{wvum ƒi•ki•i}€ƒwj•o wv •uz ~k }kjv}wxvkn vi u ~w kxvwif ~kvkkf vƒk fkw{ƒ~i}j
il ufn vƒk fkw{ƒ~i}j il  ” yv vƒwj zwkmnj u xifv}unwxvwifo uj ƒuj fkw{ƒ~i}jo ƒwmk
 ”o ƒwxƒ wj u fkw{ƒ~i} il o ƒuj p fkw{ƒ~i}j !
ivk k fkkn vƒk }kjv}wxvwif ( ’ wf i}immu}z p o jwfxk u xw}xymu} }ivuvwif il vƒk €iwfvj il
) wj nw{wvummz ƒi•ivi€wx vi vƒk wnkfvwvz if ) 
ƒk fkžv kžu•€mk jƒij u uz wf ƒwxƒ nw{wvum xifv}uxvw~wmwvz wj fiv ufumi{iyj vi xifv}uxvw~wmwvz
el 6 wj u ~iyfnkn ufn xmijkn jkv wf u ˜yxmwnkuf j€uxk vƒuv wj xifv}uxvw~mk ufn mixummz xifv}uxvw~mk
li} k|k}z €iwfv 6 ufn k|k}z fkw{ƒ~i}ƒiin il wf 6 o vƒk}k wj u fkw{ƒ~i}ƒiin @ 7 il wf
6 jyxƒ vƒuv vƒk wfxmyjwif < @ = wj ƒi•ivi€wx wf vi u xifjvufv lyfxvwif”o vƒkf 6 wj u }kv}uxv il
g i}jy–o pqs t ›ik|k}o vƒk nw{wvum ufumi{ il vƒk muvvk} }kjymv nikj fiv ƒimno wfvywvw|kmz ~kxuyjk
nw{wvum j€uxko ~kwf{ nwjx}kvko xuffiv {w|k jy xwkfvmz “fk }kjimyvwif vi •inkm ˜yxmwnkuf j€uxk li}
jyxƒ u €y}€ijk
, 344 5 ‡ 7 % ‡$  ‡
 ’ ” p ” ?$  … „…†…‡ˆ‰‰Š ‹Œ‡ ˆ‹‡… % ‰ % އ … Œ‡ ˆ ‡ ˆ‹‡ Œ> :
ŒŒ> ‚kv u €iwfv il
~k }k€}kjkfvkn ~z u}vkjwuf xii}nwfuvkj  ” ‚kv < g pt =
~k nk“fkn ~z
•už ”
wl ¡ ¡ pœ
 ” •už
 p” ” wl p i} umm  ” o  ”  ” ufn  p”  ” ev wj kujwmz jkkf vƒuv li} g pt o
jƒ}wf–j wf nw{wvummz xifvwfyiyj lujƒwif vi vƒk nw{wvum mwfk jk{•kfv ™ g ’t li}
¡ ¡ po vƒuv jƒ}wf–j ™ wf nw{wvummz xifvwfyiyj lujƒwif vi  ” li} p o ufn vƒuv
wj nw{wvummz xifvwfyiyj ev limmij vƒuv wj nw{wvummz xifv}uxvw~mk
p’
el < = k}k u nw{wvum }kv}uxvwifo vƒkf p p” iymn ~k u fkw{ƒ~i} il ~ivƒ  p”  p” ufn ’ p” ’ p” ƒk}kli}ko p p” p p” ( o u xifv}unwxvwif ƒyjo
wj fiv u nw{wvum }kv}uxv il  !
b_^ _ _ k ƒu|k kž€ufnkn if vƒk i}– il ghijkflkmno pqrst ~z nk“fwf{ nw{wvum ufumi{j il j€kxw“x vz€kj
il xifvwfyiyj lyfxvwifjo wfxmynwf{ ƒi•ki•i}€ƒwj•jo }kv}uxvwifjo kžvkfjwifjo ufn ƒi•ivi€wkj k
ƒu|k jƒif vƒuv iy} nw{wvum |k}jwifj il vƒkjk lyfxvwifj juvwjlz ufumi{j il •ufz €}i€k}vwkj ƒkmn ~z
vƒk xifvwfyiyj lyfxvwifj vƒkz •w•wxœ k ƒu|k umji jƒif €}i€k}vwkj fiv •w•wx–kn ~z iy} nw{wvum
ufumi{j il xifvwfyiyj lyfxvwifj
vƒk} }kjymvj }kmuvkn vi vƒijk il ghijkflkmno pqrst u}k €}kjkfvkn wf g u–u•y}u ufn w uuo pqqpt
_ ]c g i}jy–t i}jy–o  pqs ” ?$ Œ Š Œ> ‡ ˆ‹‡o imwjƒ ‘xwkfvw“x y~mwjƒk}jo u}ju
g—y{yfn wt —y{yfn wo  pqss” ?Œ Œ‰Œ†Šo mmzf ufn uxifo ijvif
g u–u•y}u ufn w uut u–u•y}uo o ufn  w uu pqqp” ‘i•k }kjymvj xifxk}fwf{ xiffkxvkn
ly z nw{wvum €wxvy}kjo ˆ‡‡  ‹Œ†…‡…Œ 5‡‡  p’o p
ghijkflkmnt hijkflkmno  pqrs” ifvwfyiyj lyfxvwifj if nw{wvum €wxvy}kjo ˆ‡‡  ‹Œ†…‡…Œ
5 ‡‡  o p pr 
p