✠ ✡✠☛ ☞ ✠✎
✑ ✒✓ ✔✕
✖ ✗ ✓ ✘✓
✙ ✒ ✚
✛ ✕
✖✜ ✢ ✓✣
✖ ✔✤
✙✥ ✕
✛
✓ ✒ ✦✒ ✔✓✧
✗ ✒
✘✓
✜ ✕✚★
✥ ✕✓✧
✩✪✫✫✬✭
✮✯
✰ ✱✲✳ ✬ ✴✬✫
✮ ✭
✯
✳✲
✵✶✫
✷✸ ✭✯
✳✹ ✯
✰ ✱✲
✬✺ ✴✯✮✬✺ ✻✮✪✼✯
Instituto de Biociências, Letras e Ciências Exatas
Rua Cristóvão Colombo, 2265, 15054-000
✠ ✡✠☛ ☞ ✠✎
✽✾
✿❀❁❂ ❃✿ ❄✿❅
✾❆
❇❁❂❈ ❉✿❊❂❀ ❋
❅●❁❇✿
✾ ❍✾
❀✿■
❃ ✾
❄✿❈❁ ❆
❏
●❁✿■
❑✭✪✬▲✮✯
✳✲ ✭▼
◆
❖P ◗❘ ❙
❖
❘ ❙❚❯❚❱❲
❖
❳P ❨❩❬
❙❚❯❚❱
❖P❬
❭
❪❫❴❵❛❜❫❝❞❝❵ ❡❜❢❞❝❣❞ ❤ ✐❞❣❤❫❜❢❞ ❥❦ ❧
❤❫♠ ❝❵ ♥❵❜♦❣❫❢❞ ♣❫❤q♠ r
s
❪❜❢ ❫❢ ❣❢ ♠ ❝❵
t❫♠✉❫✈❪✉❫❞❜✇ ①❵❢ ❛❞❜
❵
②❫✈❪✉❫❞❜ ❡③❞❢❞❜
②❞ ④⑤❣❜ ❝❵ ⑥ ⑦
♠ ❦♠❜⑧ ❝♠ ⑨❫♠ ✐❛❵❢ ♠
⑩
✱✲ ❶✲ ✫❷
✳✲ ❸ ✪
✲ ✵✭✬✮
✲
❾❿
➀ ➁➂➃➄➅➆➅➇➅➅
➈➉➊➋➌
➍➎ ➊➏➊
➐ ➉➑➒➌
➍ ➓➔ ➊→
➍ ➋➣
➐↔
➌➒➊➉↕➉➋➊➙
➎ ➉➏➊
➓ ➌➑➛
↔ ➌➊➙ ➜➝
➓ ➌➉
➓
➞➊➙➟➠
➍↔➍ ➋→➊
➎ ➊➙➡
➍ ➑
➐ ➊➙➢
➡➤➊
➞➊➙➟
➎
➊➥➌➊➏➋➉
➐ ➊➦
➧➙➢➑➢➨➩➫➭➯➯➢
➯➭➲➳➢➦ ➌
➓ ➢ ➵➲➭➒
↔ ➢
➸➋➌➉➑
➐➍ ➎ ➊➋➦ ➺➌
↔ ➌
➐➍ ➋➻➊
➓ ➉
➼ ➺➌
↔ ➌
➐ ➋➊
➼ ➢
➺➌➙➙➉➋
➐➍➽ ➤➊ ➾
↔ ➉➙
➐ ➋
➍ ➎
➊ ➚ ➪ ➶➑➌
➼ ➉➋➙➌
➎➍ ➎ ➉ ➝➙
➐➍ ➎➹➍ ➓ ➏
➍➹➓ ➌➙
➐➍
➘➞➴
➓ ➌➊
➎ ➉➷➉➙➬
➹ ➌
➐➍➮ ➌
➓➱ ➊✃➩❐➑➙
➐ ➌
➐ ➹➐ ➊
➎
➉❒➌➊➒➌❮➑➒➌
➍ ➙➩
➔ ➉
➐ ➋
➍
➙➉➠➌❮➑➪
➒➌
➍ ➙➝
❰➍➐➍ ➙➢
➯➢➏➊
➐ ➉➑➒➌
➍ ➓➔ ➊→
➍ ➋➣
➐↔ ➌➒➊➢
➫➢
➷➉
➎ ➌
➎➍➎ ➉➝➬
➹ ➌
➓
➣Ï➋➌➊➢➲➢➥➉➒
➹Ð ➉➋➪
➍➽ ➤➊
➎➍ ➷➉
➎ ➌
➎➍
➢ Ñ➢ Ò➙➙➌➑
➐ Ó➐ ➌➒
➍ ❐➢
➺➌
↔ ➌
➐ ➋➊
➼ ➩
➺➢➻➢ ❐❐➢ ❐➑➙
➐ ➌
➐ ➹➐ ➊
➎
➉❒➌➊➒➌❮➑➒➌
➍ ➙➩
➔ ➉
➐ ➋
➍
➙➉➠➌❮➑➒➌
➍ ➙➝
❰➍➐➍ ➙
❐❐❐➢
➈➉➊➋➌
➍➎ ➊➏➊
➐ ➉➑➒➌
➍ ➓
➔ ➊→
➍ ➋➣
➐↔
➌➒➊➉↕➉➋➊➙
➎ ➉➏➊
➓ ➌➑➛
↔ ➌➊➙➢
× Ý ÞÝßà Ý ã
å
❵♠❛❫❞ ❝♠ ✐♠❢ ❵❪✉❫❞ ❤ ①♠æ❞❛ç❢④❫✉♠ ❵ è❵❛♠❜ ❝❵ ✐♠❤❫❪é④❫♠❜
✩✪✫✫✬✭✮✯
✰ ✱✲ ✯
ê
✭✬✫✬▲✮✯
✳ ✯
ê ✯✭✯
✲ë ✮✬▲
✰ ✱✲✳✲ ✮ ì✮
✹í✲
✳
✬ ✴✬✫
✮
✭✬ ✬✺ ✴
✯✮ ✬✺ ✻
✮ ✪✼
✯î ✻✭✬
✯
✳
✬ ï▲ ✻
í ✪✫✬
ï
êí ✪✼✯
✳ ✯î ð
✹ ▲✮
✲ ✯
✲ ñ ▲✫✮✪✮
✹ ✮
✲ ✳ ✬ ò✪
✲
✼✪ó▲✼✪✯✫î
ô✬✮✭✯✫ ✬ õ✪ó▲✼✪✯✫ ö
÷ ✯✮✯✫
✳ ✯
ø▲✪
ù ✬✭✫✪
✳ ✯
✳ ✬
ö✫✮✯
✳✹ ✯
í ✵✯
✹í ✪✫✮✯
ú
❶ û
í ✪
✲ ✳
✬ ✴✬✫ü
✹ ✪✮✯ ý✪
íþ✲ ÿî
õ
✯ ✺
ê✹ ✫
✳ ✬ ⑩
✱✲ ❶✲ ✫❷
✳✲ ❸ ✪
✲ ✵✭✬
✮
✲
t❞❪✉❞ ❡③❞ ④❫❪❞❝♠❛❞
✵✭
✲ ✁
✩✭ ✩✪✺✪✮✯✭ ✂
✲í ✬
ù
✩✪✺✪✮✭
✲ù
✵✭
✲ ✁✬✫✫
✲ ✭✄✪
✮
✹í ✯
✭
ø☎ö⑩✵
✷ ⑩
✱✲ ❶✲ ✫❷
✳✲❸ ✪
✲ ✵✭✬✮
✲
❑✭✪✬▲✮✯
✳✲ ✭
✵✭
✲ ✁ ✩✭
✆ ✯
í✳
✪✭ ï▲✮ ✝▲✪
✲
✴✬▲✬✞✯✮✮
✲
✵✭
✲ ✁✬✫✫
✲ ✭✄✪✮
✹í ✯✭
ø ⑩✵
✷ ⑩
✱✲ õ✯✭
í✲ ✫
✵✭
✲
✁ ✩✭ ï
í
✪ ✴✬✫✫✯
✲✹✳ ✪
✵✭
✲ ✁✬✫✫
✲ ✭ ï
✳ ð
✹ ▲✮
✲
ø☎ ö⑩✵
✷ ⑩
✱✲ ❶✲ ✫❷
✳✲❸ ✪
✲ ✵✭✬
✮
✲
⑩ ✫❷ ✪ ✵✭✬✮
î
ï
✲ ✫ ✺✬
✹ ✫ ✯✺✯
✳✲ ✫
ê ✯✪✫î
❶✲ ✫❷✬
✳ ✻
í ✪✯
ï
✲ ✫ ✺✬
✹ ✫ ✪✭✺
✱✲ ✫îö
✳ ✺✪
í ✮
✲ ▲î
❶✲ ✫❷
í
✪✯î✟✭✪✼✯✬ ö
í ✪✁
✯
✠
ï
✲ ✫ ✺✬
✹ ✫ ü
✹ ✬✭✪
✳✲
✫ ✯✺✪✞
✲ ✫
✡☛☞✌✍
✒✓✔✕ ✖✗ ✖✚✛✜✢ ï ✩✬ ✹ ✫î ê ✬ í ✯✫ ✲ê✲ ✭✮ ✹ ▲✪ ✳ ✯ ✳ ✬✫ ✬ ê ✬✫✫ ✲ ✯✫✬▲✼✯▲✮✯ ✳✲ ✭✯✫ü ✹ ✬ ✬▲✼ ✲ ▲✮✭ ✲ ✬✺✺✬ ✹ ✼✯✺✪▲ þ✲ ï ✺✪▲ þ ✯ ✁✯✺ì í ✪✯î ê ✬ í✲ ✯ ê✲ ✪ ✲ î ✼ ✲ ✺ ê ✭✬✬▲✫ ✱✲ ✬ ✪▲✼✬▲✮✪ ù✲ ✼ ✲ ▲✮ ì▲ ✹✲ ê ✯✭✯ ✭✬✯ í ✪ ✠ ✯ ✰ ✱✲ ✳ ✬ ✺✯✪✫ ✹ ✺✯✼ ✲ ▲ü ✹ ✪✫✮✯ ï ✲ ✺✬ ✹✲ ✭✪✬▲✮✯ ✳✲ ✭î✵✭ ✲ ✁ ✩✭ ✩✪✺✪✮✯✭ ✂ ✲í ✬ ù ✩✪✺✪✮✭ ✲ù î ê ✬ í✲ ✯ ê✲ ✪ ✲ î ✯✺✪ ✠ ✯ ✳ ✬ ✬ ✯ ✹÷ ì í ✪ ✲ ✳✹ ✭✯▲✮✬ ✯ ✲ ✭✪✬▲✮✯ ✰ ✱✲ ✳ ✬✫✮✬ ✮✭✯ ë ✯ íþ✲ ï ✲ ✫ ê ✭ ✲ ✁✬✫✫ ✲ ✭✬✫î ✵✭ ✲
✁ ✩✭ ï
í
✪ ✴✬✫✫✯
✲✹✳ ✪ ✬ ✵✭
✲
✁ ✩✭ ï
í ✯✞✯✼ ✲ ▲✬ ⑩✭✪ ❸ ✯▲✞✯î ê ✬ í ✯✫ ✪▲û✺✬✭✯✫ ✬ ù ✯ í ✪ ✲ ✫✯✫ ✫ ✹ ✞✬✫✮ ✣✬✫ ê ✯✭✯✬✫✮✯ ✳ ✪✫✫✬✭✮✯ ✰ ✱✲ ï ✴ ✯ ✭✪✫ ✯ ✳ ✬ ⑩ ✲✹✠ ✯ õ ✲ ✫ ✮✯ ✬ ❸ ✬✞✪▲ ✯ ✳ ✯ ⑩✪ íù ✯ ô ✯ ✺ ëí ❷✺ î ê ✬ í ✯ ✫✼ ✲ ▲ ✮ ✭✪ ë✹ ✪ ✰ ✣ ✬✫ ✯ ✲ ê ✭✬✫✬▲ ✮ ✬ ✮✭✯ ë ✯ íþ✲ ï ✲ ✫ ê ✭ ✲ ✁✬✫✫ ✲ ✭✬✫ ✳ ✬ ✞✭✯ ✳✹ ✯ ✰ ✱✲ ✬ ê ✶✫ ✷ ✞✭✯ ✳✹ ✯ ✰ ✱✲ î ✬✺ ✬✫ ê ✬✼✪✯ í ✤ ✵✭ ✲ ✁❾ ✩✭❾ ✴✯✭✪✯ ✸ ✲
✭✬✮✬õ✯✭✭✬✭✯ï▲
✳ ✭✯ ✳ ✬î✵✭ ✲ ✁ ✩✭ õ í ✻ ✹✳ ✪ ✲ ï✞ ✹ ✪▲✯ í✳✲ ò ✹✠✠ ✪ ✤ ✵✭ ✲ ✁❾ ✩✭❾ ö í ✪✯▲✯ ✥✯ ù ✪✬✭ ô✪▲ þ ✯✭✬✫ ✳ ✬ ï▲ ✳ ✭✯ ✳ ✬ ✬ ✯ ✲ ✵✭ ✲
✁ ✩✭ ï
í ✯✞✯✼ ✲ ▲✬ ⑩✭✪ ❸ ✯▲✞✯î ê ✬ í ✯ ✳ ✬ ✳ ✪✼✯ ✰ ✱✲ î ✯✮✬▲ ✰ ✱✲ ✬ ù ✯ í ✪ ✲ ✫ ✲ ✫✬▲✫✪▲✯✺✬▲✮ ✲ ✫ ï ✮ ✲✳✲ ✫ ✲ ✫ ✼ ✲í ✬✞✯✫ ✳ ✬ ✞✭✯ ✳✹ ✯ ✰ ✱✲ î ✬✺ ✬✫ ê ✬✼✪✯ í ✯ ✲ ✫ ✯✺✪✞ ✲ ✫ ✩✯▲✪ í✲ î ô✬ ✲ ▲✯✭ ✳✲ î ❸ ✯✁✯✬ í î ❸✲✳ ✭✪✞ ✲ î ô✬✮✪✼✪✯î ✴✯✭✼ ✲ ✫ ✦ ✪ íí ✪✯✺î ✄ þ ✯✪✫✯î ❸ ✬▲✯✮✯ ❸✲
✫✪✺✬✪✭✬ ✬ ï▲
✳ ✭❷✪✯î ê ✬ í ✯ ✯✺✪ ✠ ✯ ✳ ✬ ✳✹ ✭✯▲✮✬ ü ✹ ✯✮✭ ✲ ✯▲ ✲ ✫ ✳ ✬✼ ✲ ▲ ù ì ù ✪ ✲ ï ✲ ✫ ✯✺✪✞ ✲ ✫ ✳ ✬ ê ✶✫ ✷ ✞✭✯ ✳✹ ✯ ✰ ✱✲ ï í✧ ▲✬ î õ✪▲✮ ✧ ✯î
õ✭✪✫✮✪✯▲✬ ✴✯✭✪✯î ö
✳ ✲ ê í ✯
✲ ✲ í✧
ý
✳✲ î
ë✲ î
✸
✹íþ þ íí
î
ê ✬
í✲ ✫✺
✲ ✺✬▲✮
✲ ✫
✳ ✬✬✫✮
✹✳✲ ✬
✳ ✪✫✮✭✯
✰ ✱✲ ▲✯ú✫✯
í ✪▲
þ
✯ÿ ✬ú✼✯✁✬
✠ ✪▲
þ✲ ✫ÿ
✯
ê ✭
✲ù ✬✪✮✯
✳✲ ✫▲
✲
✩õõöî
✯ ▲✞û✫
✮ ✪
✯ ✫
✳ ✪
ù ✪
✳ ✪
✳ ✯
✫
î ✭✪✫
✲ ✫✫
✲ ✺
✯
✳✲ ✫
î ✞✭
✯ ▲
✳ ✬
✯ ✺✪
✠ ✯
✳ ✬✬
✹ ✺
✯
ê ✭✬✼✪
✲ ✫
✯ ✼
✲í ✯
ë✲ ✭
✯
✰ ✱✲✤ ✭✬
✯
í ✪
✠ ✯
✷
✰ ✱✲✳
✬✫✮✬ ✮✭✯
ë ✯
íþ✲
ï õ
í ❷✼✪✯
ê ✬
í✲ ✯
ê✲ ✪
✲ ✬ ✼
✲ ✺
ê ✯▲
þ ✬✪✭✪✫✺
✲✳✲ ✫ û
í ✮✪✺
✲
✫ ✺✬✫✬✫
ï
✮
✲✳✲ ✫
✲ ✫ ✼
✲í ✬✞✯✫î
ê ✬✫✫
✲ ✯✫ ✬ ✁
✹ ▲✼✪
✲ ▲ ✻✭✪
✲ ✫
✳✲ ñ ò
ñ
ôõö ü
✹ ✬î
✳ ✪✭✬✮✯
✲✹ ✪▲
✳
✪✭✬✮✯✺✬▲✮✬î
✼
✲ ▲✮✭✪
ë✹ ì✭✯✺
ê ✯✭✯
✯ ✬
í ✯
ë✲ ✭✯
✰ ✱✲ ✳
✬✫✮✬ ✮✭✯
ë ✯
íþ✲
ö✺ ✬✫
ê ✬✼✪✯
í ✯
✲ ✫
✁
✹ ▲✼✪
✲ ▲ ✻✭✪
✲ ✫
✳✲
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✲ ✫
✳ ✬
ê✲ ✮✬▲✼✪✯
í
✬ ✬▲✬✭✞✪✯
í✲
✞✯✭ì✮✺✪✼
✲ ✫î ✺✬
✳ ✪
✳ ✯
✳ ✬
✬ü
✹ ✪
í ì
ë ✭✪
✲ ✬ ✫
✹ ✯✫
ê ✭
✲ê ✭✪✬
✳ ✯
✳ ✬✫ ✫
✱✲ ✁
✲ ✭▲✬✼✪
✳ ✯✫ ▲
✲ ✫✬✞
✹ ▲
✳✲ ✼✯
ê ì✮
✹í✲ ❑
ü
✹ ✯✭✮
✲ ✼✯
ê ì✮
✹í✲
✳ ✯
✳ ✪✫✫✬✭✮✯
✰ ✱✲ ✼
✲
▲✮❷✺ ✭✬✫
✹í ✮✯
✳✲ ✫ ü
✹
✬ ✼✯✭✯✼✮✬✭✪
✠ ✯✺
✲ ✫
✹ê✲ ✭✮✬
✳ ✯
✺✬
✳ ✪
✳ ✯
✳ ✬ ✬ü
✹ ✪
í ì
ë ✭✪
✲
ý✪▲✯
í
✺✬▲✮✬î▲
✲ û
í ✮✪✺
✲ ✼✯
ê ì✮
✹í✲ î
✁
✲
✭▲✬✼✬✺
✲ ✫
✯
í ✞
✹ ▲✫✬
÷ ✬✺
êí✲ ✫
✲ ▲
✳ ✬
✳
✬✮✬✭✺✪▲✯✺
✲ ✫✬
÷êí ✪✼✪
✷
✮✯✺✬▲✮✬
✯ ✺✬
✳ ✪
✳ ✯
✳ ✬✬ü
✹ ✪
í ì
ë ✭✪
✲ ý
✲
✭▲✬✼✬✺
✲ ✫
✯
✳ ✬✺
✲ ▲✫✮✭✯
✰ ✱✲✳ ✬
✹ ✺✭✬✫
✹í ✮✯
✳✲
✪▲✮✬✭✬✫✫✯▲✮✬
✫ ✭✬
✯
✼ ▲ ✬✭✞ó▲✼✪✯
✯ ✺✬ ✪
✯
❵ ❛
✓✜❜✓ ✘
✖✕✔✕ ✖✢ ❝✖✓✔
✘
✢ ✕ ✖ ❞✯✚❪ ❡✖✢
❢
✜ ✙
❜❣✖❤✔✢
☎✬✫✮✬ ✼✯
ê ì✮
✹í✲ ✪▲✮✭
✲✳✹✠ ✪✺
✲ ✫ ✼
✲ ▲✼✬✪✮
✲ ✫
ë ✻✫✪✼
✲ ✫
✳ ✯
✄✬
✲ ✭✪✯
✳ ✯
✴✬
✳ ✪
✳ ✯
✬ ï▲ ✻
í ✪✫✬
ý
✹ ▲
✷
✼✪
✲ ▲✯
í
✄✯✪✫✭✬✫
✹í ✮✯
✳✲ ✫✫
✱✲ ✬▲✼
✲ ▲✮✭✯
✳✲ ✫
ê✲ ✭✬
÷ ✬✺
êí✲
✬✺ ✐❹❺ ❥
✬ ✐❹❈ ❥
❦❧♠ ♥
♦ ♣ q
rs t ✉
✈ q
✉✇
q ①②①
♦
✩
✹ ✭✯▲✮✬
✮
✲✳✲ ✲ ▲
✲ ✫✫
✲ ✮✭✯
ë ✯
íþ✲
✬✫✮✯✭✬✺
✲
✫ ✮✭✯✮✯▲
✳✲ ✳ ✬ ✁
✹ ▲
✰
✣✬✫ ✬✺
C
:=
C
S
{∞}
î ✼✲ ▲
þ ✬✼✪
✳ ✯
✮✯✺
ë ❷✺ ✼
✲
✺ ✬✫✁✬✭✯
✳ ✬
❸
✪✬✺✯▲▲
ê ✬
í✲ ✁✯✮
✲ ✳ ✬
✲ ✼
✲ ▲ð
✹ ▲✮
✲ ✫✬✭
þ✲ ✺✬
✲ ✺
✲ ✭✁
✲ ✯
✬✫✁✬✭✯
S
2
î þ✲✺✬
✲ ✺
✲ ✭
✬ ✫✺
✲ ✳ ✯
✳✲ ê ✬
í ✯
ê ✭
✲ ð
✬
✰ ✱✲
✬✫✮✬✭✬
✲ ✞✭ ✻
✬ ✼✯ ✵
✲ ✭✮✯▲✮
✲
C
❷ ✹✺ ✬✫
ê ✯
✰✲
✮
✲ê✲í ✶✞✪✼
✲ ✼
✲ ✺
✯✮
✲ê✲í✲ ✞✪✯✪▲
✳✹✠ ✪
✳ ✯
ê ✬
í✲þ✲ ✺✬
✲ ✺
✲ ✭
✬ ✫✺
✲
③
× ❳
ã
Ö❀❁Ù ❂④ ✽④ ⑤☛⑥⑦
f
:
X
→
R
⑧ ✍✎⑨
X
⊆
C
⑧ ☞☛⑩❶✌
⑨✎❷ ✎ ❸ ✌
⑨ ✌❹☛
✌❶❺
☛
❻ ✌
✎❻ ☞⑦
❺❼❶ ❽ ❾
✎
f
✍
✎⑨✎❷ ☛❶☞
✎
lim inf
x
→
z0
f
(x) = lim
r
→0
{
x
∈
D
inf
r(
z0
)∩
X
f
(x)
}
,
✎ ❶☞☛
D
r
(z0) =
{
z
:
|
z
|
< r
}
.
③
× ❳
ã
Ö❀❁Ù ❂④❂④ ⑤☛⑥⑦
f
:
K
→
R
❼
⑨ ⑦
❺❼❶ ❽ ❾
✎ ☞☛
⑩❶✌☞⑦
☛
⑨ ❼
⑨ ✍
✎ ❶
⑥ ❼❶❹
✎
K
⊂
C
✏ ❿ ❶❹❾✎
⑧
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹✌❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❼
⑨ ➂✎ ❶❹
✎
x
∈
K
⑧ ☛⑨ ➃ ❼☛
α < f
(x)
⑧ ❷☛ ☛
➄ ✌
❷ ❹☛ ❼
⑨ ⑦
➅ ✌
➆
✌❶➇⑦❶ ❽⑦
U
☞☛x
❹⑦ ❸ ➃❼☛
α < f
(y)
➂ ⑦❻ ⑦ ❹
✎ ☞
✎
y
∈
U
T
K
✏ ✡✌➆ ☛
⑨✎❷ ➃ ❼☛
f
➀ ❷☛
⑨ ✌➁
✍
✎
❶❹➈❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨
K
❷ ☛ ☛❸ ⑦
➀ ❷
☛
⑨ ✌➁✍
✎
❶❹➈❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❹
✎ ☞
✎➂✎ ❶❹
✎ ☞☛ ➉
✏
➊
⑨ ⑦
❺❼❶ ❽ ❾
✎
f
➀ ❷☛
⑨ ✌➁✍
✎ ❶❹✌❶❼⑦
❷ ❼
➂ ☛
❻ ✌
✎❻⑨ ☛❶❹☛
☛
⑨
K
❷ ☛−
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹✌❶❼⑦✌❶❺
☛
❻ ✌
✎❻⑨ ☛❶❹☛
☛
⑨
K
✏ ➋× Ù
ß× ÞÝ ❂④ ✽④
⑤☛
⑥⑦
f
❼ ⑨⑦ ❺❼❶ ❽ ❾
✎
☞☛⑩❶✌☞⑦ ☛
⑨ ❼
⑨ ☞✌
❷ ✍
✎
D
⑧ ☛❶❹❾✎ ➅ ⑦
❸ ☛
⑨ ⑦
❷ ❷ ☛
➌ ❼✌❶❹☛
❷
☛
➃ ❼✌
➅ ⑦
❸➍ ❶✍✌⑦
❷➎
(1)
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹✌❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨
D
➏(2)
➐ ⑦❻ ⑦
✍⑦☞⑦
z0
∈
D
⑧ ❶➑❷ ❹☛
⑨✎❷
f
(z0)
≤
lim inf
z
→
z0
f
(z)
↔↕➙➛➜➝ ➞↕➝➟↕➠➡➠
(3)
➐ ⑦❻ ⑦ ✍⑦☞⑦
α
∈
R
⑧ ✎✍
✎ ❶
⑥ ❼❶❹
✎
{
z
∈
D
:
f(z)
> α
}
➀ ⑦➤☛❻ ❹
✎
➏
③
× Þ Ù
ã Úä
ßÝ ❀❁Ù
✵
✵✭✪✺✬✪✭✯✺✬▲✮✬
ù ✯✺
✲ ✫
ê ✭
✲ù ✯✭ ü
✹ ✬
(1)
⇐⇒
(2)
⑩
✹ê✲ ▲
þ ✯✺
✲ ✫ü
✹ ✬
✁
❷✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬✬✺
D
⑩ ✯ë ✬✺
✲ ✫ü
✹ ✬
f
(z
0
)
> f
(z
0
)
−
ε
ê ✯✭✯
✮
✲✳✲
ε >
0
✵ ✲✭✮✯▲✮
✲ ê ✬
í ✯
✫✬✺✪
✷ ✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳ ✬
✪▲✁✬✭✪
✲ ✭
✳ ✬
f
✬✺D
ê✲✳ ✬✺✲
✫ ✞✯✭✯▲✮✪✭
ü
✹ ✬ ✬
÷ ✪✫✮✬
✹ ✺✯
ù ✪
✠ ✪▲
þ ✯▲
✰ ✯
U
z0
✮✯
í ü
✹ ✬
f
(y)
> f
(z0)
−
ε
ê ✯✭✯✮
✲✳✲
y
∈
U
z0
ö▲✮
✱✲ î
✬
÷ ✪✫✮✬
✹ ✺
δ
✮✯ íü
✹ ✬
ê ✯✭✯
✮
✲✳✲
z
∈
D
δ
(z0)
⊂
U
z0
î
✲ ▲
✳ ✬
D
δ
(z0
) :=
{
z
:
|
z
|
< δ
}
î ✮✬✺
✲ ✫ ü
✹ ✬
inf
z
∈
D
δ(
z0
)
f
(z)
≥
f
(z0)
−
ε.
ï✫✫✪✺î✫✯
ë ✬▲
✳✲ ü
✹ ✬
lim inf
z
→
z0
f
(z) = lim
r
→0
z
∈
D
inf
δ(
z0
)
f
(z)
î
✲ë ✮✬✺
✲ ✫
lim inf
z
→
z0
f(z)
≥
f(z0
)
−
ε.
ý✯
✠ ✬▲
✳✲
ε
→
0
î ✲ë✮✬✺
✲ ✫
lim inf
z
→
z0
f(z)
≥
f(z0
).
✵✯✭✯
ê ✭
✲ù ✯✭✺
✲ ✫
✯ ✪✺
êí ✪✼✯
✰ ✱✲ ✼
✲
▲✮✭ ✻✭✪✯î✫
✹ê✲ ▲
þ ✯✺
✲ ✫ü
✹ ✬
f
✫✯✮✪✫✁ ✯✰ ✯
(2)
î ✺✯✫ü✹ ✬▲
✱✲
✫✬
ð✯ ✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬✬✺
D
ö▲✮ ✱✲î ✬
÷ ✪✫✮✬✺
z
∈
D
✬ε >
0
î✮✯ íü
✹ ✬î
ê ✯✭✯
✮
✲✳✲
δ
î ✮✬✺✲ ✫
f
(z)
> f
(z)
−
ε
ê✲ ✭❷✺f(z)
≤
f
(z)
−
ε
ê ✯✭✯✮
✲✳✲
z
∈
D
δ
(z)
ï✫✫✪✺î✫✯
ë ✬▲
✳✲
ü
✹ ✬
lim inf
z
→
z
f
(z) = lim
δ
→0
D
inf
δ(
z
)
f(z),
✮ ✬✺
✲ ✫ ü
✹ ✬
lim inf
z
→
z
f(z)
≤
f
(z)
−
ε
ý✯
✠ ✬▲
✳✲
ε
→
0
î ✼✲ ▲✼
í✹ ì✺
✲ ✫ü
✹ ✬
lim inf
z
→
z
f(z)
≤
f(z)
✲ ü
✹ ✬✼
✲
▲✮✭ ✻✭✪✯▲
✲ ✫✫✯
þ ✪
ê
✶✮✬✫✬ ✵
✲ ✭✮✯▲✮
✲ î
✯✫
þ ✪
ê ✶✮✬✫✬✫
✳ ✬✫✬✺✪
✷ ✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳ ✬
✪▲✁✬✭✪
✲ ✭ ✬✫✮
✱✲
✫✯✮✪✫✁✬✪✮✯✫î
✳ ✬
✲ ▲
✳ ✬ ✼
✲ ▲✼
í✹ ì✺
✲ ✫ ü
✹ ✬
f
❷ ✫✬✺✪ ✷✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬✬✺
D
ï✞
✲ ✭✯
ù ✯✺
✲ ✫
ê ✭
✲ù ✯✭ ü
✹ ✬
(1)
⇐⇒
(3)
⑩
✹ê✲ ▲
þ ✯✺
✲ ✫ ü
✹ ✬
f
✫✬ð ✯✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬ ✬✺
D
✆✯✺
✲ ✫
ê ✭
✲ù ✯✭ ü
✹ ✬
✲
✼
✲ ▲
ð
✹ ▲✮
✲
A
:=
{
z
∈
D
:
f
(z)
> α
}
❷ ✯ë ✬✭✮
✲ î
✫✬
ð✯
z
∈
A
î ✬▲✮✱✲
f(z)
> α
î í✲✞
✲ ê ✬
í ✯
✳ ✬
✬ ▲✪
✰ ✱✲ ✳ ✬ ✫✬✺✪
✷ ✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳
✬ ✪▲✁✬✭✪
✲ ✭ ✬
÷ ✪✫✮✬
✹ ✺✯
ù ✪
✠ ✪▲
þ ✯▲
✰ ✯ ✯
ë ✬✭✮✯
U
z
î ✬✺
✮
✲ ✭▲
✲ ✳ ✬
z
î ✮✯í ü
✹ ✬
f
(z)
> α
ê ✯✭✯✮
✲✳✲
z
∈
U
z
î
✲✹ ✫✬
ð✯î
U
z
⊂
A
✵
✲ ✭✮✯▲✮
✲ î
A
❷ ✯ë ✬✭✮
✲
❸ ✬✼✪
ê ✭
✲
✼✯✺✬▲✮✬î✫✬ð
✯
z
∈
D
✹ ✺ê✲ ▲✮
✲ ✮✯
í ü
✹ ✬
f
(z)
> α
✪✫✮ ✲✪✺
êí ✪✼✯ ü
✹ ✬
✲ ✼
✲ ▲
ð
✹ ▲✮
✲
A
:=
{
z
∈
D
:
f(z)
> α
}
❷ ▲ ✱✲ù ✯
✠ ✪
✲ î
✼
✲ ✺
✲ ê✲ ✭
þ ✪
ê
✶✮✬✫✬ ✬
í ✬ ❷
✯
ë ✬✭✮
✲ î
ë ✯✫✮✯
✮
✲ ✺✯✭✺
✲ ✫
U
=
A
▲✯ ✳✬
✬ ▲✪
✰ ✱✲✳ ✬✫✬✺✪
✷ ✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳ ✬
✪▲✁✬✭✪
✲ ✭
ê ✯✭✯✼
✲ ▲✼
í✹ ✪✭✺
✲ ✫ü
✹ ✬
f
❷✫✬✺✪ ✷✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬▲
✲ ê✲ ▲✮
✲
z
➋
× Ù
ß× ÞÝ ❂④❂④
⑤☛
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹✌❶❼⑦✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❼
⑨ ✍
✎ ❶⑥❼❶❹
✎ ✍
✎⑨➂ ⑦ ✍❹
✎
K
⑧ ☛❶❹❾✎
f
⑦❹✌❶ ➌☛
✎⑨ ➈❶✌
⑨✎ ☛
⑨
↔↕➙➛➜➝ ➞↕➝➟↕➠➡➠
③
× Þ Ù
ã Úä
ßÝ ❀❁Ù
✵ ✵
✲ ✭
f
✫✬✭ ✫✬✺✪ ✷✼
✲ ▲✮✪▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬ ✬✺
✹ ✺ ✼
✲ ▲
ð
✹ ▲✮
✲ ✼
✲ ✺
ê ✯✼✮
✲
K
î ✫✬✞✹ ✬
✳✲ ✄✬
✲
✭✬✺✯ ❻❹ ✪✮✬✺
(3)
î ✲✫ ✼
✲ ▲
ð
✹ ▲✮
✲ ✫
U
n
:=
{
z
∈
K
:
f
(z)
>
−
n
}
, n
≥
1
✫
✱✲
✯
ë ✬✭
✮
✲ ✫
î ✬✬✫
✮ ✻✼
í ✯
✭
✲ ü
✹ ✬✬
í ✬✫✁
✲ ✭✺
✯ ✺
✹ ✺
✯ ✼
✲ë ✬✭
✮
✹ ✭
✯
✳ ✬
K
⑩✬▲ ✳✲K
✼ ✲✺
ê ✯
✼
✮
✲ ✬
í ✬
✯
✳ ✺✪
✮ ✬
✫
✹ë
✷ ✼
✲ë ✬✭✮
✹ ✭✯
✬ ▲✪✮✯
✵
✲ ✭✮✯▲✮
✲ î
f
❷ í✪✺✪✮✯
✳ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬✬✺
K
⑩✬
ð✯
M
= inf
K
f
❑✫
✼
✲ ▲ð
✹ ▲✮
✲ ✫
V
n
:=
{
z
∈
K
:
f
(z)
> M
+ 1/n
}
, n
≥
1
▲
✱✲ê✲✳ ✬✺✼
✲ë ✭✪✭
K
✩✬ ✁
✯✮
✲ î
✼✯✫
✲ ✬
í ✬✫ ✼
✲ë
✭✪✫✫✬✺✬▲✮
✱✲ ✬
í
✬✫✮✯✺
ë ❷✺
✯
✳
✺✪✮✪✭✪✯✺
✹ ✺✯✫
✹ë
✷ ✼
✲ë ✬✭✮
✹ ✭✯
✬ ▲✪✮✯î
✲✹ ✫✬
ð✯î ✬
÷
✪✫✮✪✭✪✯✺
n
j
, j
= 1, . . . , k
✮✯
í ü
✹ ✬
✯ ✁
✯✺ì
í ✪✯
✳✲ ✫ ✼
✲ ▲
ð
✹ ▲✮
✲ ✫
{
V
n
j
}
k
j
=1
✼
✲ë ✭✪✭✪✯✺
K
î ✪✫✮✲
✯✼✯✭✭✬✮✯ ü
✹ ✬
ê ✯✭✯
m > n
k
✲ ✫ ✼
✲ ▲
ð
✹ ▲✮
✲ ✫
V
m
=
V
n
k
î ✯✫✫✪✺
f(z)
> M
+ 1/n
k
ê ✯✭✯
✮
✲✳✲
z
✬ ê✯✭✯
✮
✲✳✲
m > n
k
î
ê✲ ✭✮✯▲✮
✲
inf
K
f(z)
≥
M
+ 1/m > M
î
✲ ü
✹ ✬ ❷
✹ ✺
✯
ë ✫
✹ ✭
✳✲ ✳ ✬
✯✼
✲ ✭
✳✲ ✼
✲ ✺
✯
✳ ✬
✬ ▲✪
✰ ✱✲✳ ✬
M
õ✲ ▲✫✬ü
✹
✬▲✮✬✺✬▲✮✬î ✬
÷ ✪✫✮✬
z
✮✯ íü
✹ ✬
M
≤
f(z)
≤
M
+ 1/n
ê ✯✭✯
✮
✲✳✲
n >
1
î ✲✹✫✬
ð✯î
u(z) =
M
ê ✯✭✯ê ✬
í✲ ✺✬▲
✲ ✫
✹ ✺
z
∈
K.
➋
× Ù
ß× ÞÝ ❂④●④
⑤☛
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹✌❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❼
⑨ ✍
✎ ❶⑥❼❶❹
✎ ✍
✎⑨➂ ⑦ ✍❹
✎
K
❷ ☛⑧ ☛
❷✎⑨ ☛❶❹☛
❷ ☛
⑧ ☛
❸ ⑦
➀ ❸
✌
⑨ ✌❹☛
➂✎ ❶❹❼⑦
❸ ☞☛ ❼
⑨ ❷ ☛
➃ ❼
➍
❶✍✌⑦❺❼❶ ❽➦☛
❷ ✍
❻ ☛
❷ ✍☛❶❹☛
❷ ☛
✍
✎ ❶❹✌❶❼⑦
❷
✏
③
× Þ Ù
ã Úä
ßÝ ❀❁Ù
✵ ✆ ✯✺
✲ ✫
ê ✭
✲ù ✯✭
ê
✭✪✺✬✪✭✯✺✬▲✮✬
✯ ✫
✹✬
✼✪ó▲✼✪✯ ⑩
✹ê✲ ▲
þ ✯✺
✲ ✫ ü
✹ ✬
f
✫✬ ð✯í ✪✺✪✮✬
ê✲ ▲✮
✹ ✯
í✳ ✬
✹
✺✯✫✬ü
✹ ó▲✼✪✯
✳ ✬
✁
✹ ▲
✰ ✣✬✫✼
✲ ▲✮ ì▲
✹
✯✫✬✼✭✬✫✼✬▲✮✬✫
f
n
ö▲✮
✱✲ î
ê ✯✭✯
y
∈
K
✮✯
í ü
✹ ✬
f
(y)
> α
✪✺ êí✪✼✯ ü
✹ ✬
lim
f
n
(y) =
f
(y)
> α
✬
f
n
✫
✱✲ ✁
✹ ▲
✰ ✣✬✫ ✼
✲ ▲✮✪▲
✹ ✯✫î ✬
÷ ✪✫✮✬✺
δ >
0
✬ε >
0
î✮✯ íü
✹ ✬î
ê ✯✭✯
|
x
−
y
|
< δ
✳ ✬ù ✬✺
✲ ✫ ✮✬✭
f
n
(x)
> α
+
ε
î ✁
✯
✠ ✬▲
✳✲
n
→
+
∞
î ✲ë✮✬✺
✲ ✫ ü
✹ ✬
f
(x)
≥
α
ê ✯✭✯✮
✲✳✲
x
✮✯ íü
✹ ✬
|
x
−
y
|
< δ
✵ ✲✭✮✯▲✮
✲ î
✫✬✞
✹ ✬ ü
✹ ✬
f
❷ ✫✬✺✪ ✷✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬ ✬✺
K
❸ ✬✼✪
ê ✭
✲ ✼
✯ ✺✬▲
✮ ✬
î ✫✬
ð✯
f
✹ ✺✯ ✁
✹ ▲
✰ ✱✲ ✫✬✺✪
✷ ✼
✲ ▲
✮ ì ▲
✹ ✯
✪▲✁✬✭✪
✲ ✭✺✬▲
✮ ✬ ▲
✱✲ ✪
✳ ✬▲
✮ ✪✼
✯ ✺✬▲
✮ ✬
✪✞
✹ ✯
í ✯
+
∞
✄ ✲✺✬✺
✲ ✫
✯ ✫✬ü
✹ ó▲✼✪✯
✳ ✬
✁
✹ ▲
✰ ✣✬✫
f
n
(x) := inf
y
∈
K
{
f
(y) +
n
|
x
−
y
|}
.
ö✫✮✯✫✁
✹ ▲
✰ ✣✬✫
f
n
✬✫✮
✱✲ ë ✬✺
✳ ✬
✬ ▲✪
✳ ✯✫
✹ ✺✯
ù ✬
✠ ü
✹ ✬
K
❷✼ ✲✺
ê ✯✼✮
✲
✆ ✯✺
✲ ✫
ê ✭
✲ù ✯✭ ü
✹ ✬
ê ✯✭✯ ü
✹ ✯✪✫ü
✹ ✬✭
x, x0
∈
K
î ✮✬✺✲ ✫
↔↕➙➛➜➝ ➞↕➝➟↕➠➡➠
✩✬ ✁
✯✮
✲ î
f
n
(x) = inf
y
∈
K
{
f
(y) +
n
|
x
−
y
|}
= inf
y
∈
K
{
f
(y) +
n
|
x
−
x0
+
x0
−
y
|}
≤
inf
y
∈
K
{
f
(y) +
n
|
x
−
x0
|
+
n
|
x0
−
y
|}
≤
inf
y
∈
K
{
f
(y) +
n
|
x
0
−
y
|}
+
n
|
x
−
x
0
|
=
f
n
(x
0
) +
n
|
x
−
x
0
|
,
✬✯▲✯
í✲
✞✯✺✬▲✮✬
ê✲✳ ✬✺
✲ ✫
ê ✭
✲ù ✯✭ü
✹ ✬
f
n
(x)
≥
f
n
(x0)
−
n
|
x
−
x0
|
✩✬✫✮✯ ✁
✲ ✭✺✯î
✳ ✯
✳✲
0
< ε
ë ✯✫✮✯
✮
✲ ✺✯✭✺
✲ ✫
✹ ✺
δ
=
ε/n
✬ ✼ ✲▲✼
í✹ ✪✭✬✺
✲ ✫ ü
✹ ✬
f
n
❷ ✼
✲ ▲✮ ì▲
✹ ✯
ö
f
n
(x)
< f
n
+1(x)
ê ✯✭✯
✮
✲✳✲
x
✫✬✞ ✹✬
✳
✪✭✬✮✯✺✬▲✮✬
✳ ✯✫
ê ✭
✲ê ✭✪✬
✳ ✯
✳ ✬✫
✳ ✬ ì▲
✬ ✺
✲✳ ✬✁
✹ ▲
✰ ✣✬✫
✵✭✬✼✪✫✯✺
✲ ✫ ✯✞
✲ ✭✯
ê ✭
✲ù ✯✭
✯
ê ✬▲✯✫ ü
✹ ✬
lim
f
n
(x) =
f(x).
✵✯✭✯ ✮✯▲✮
✲ ✁✯
✠ ✬▲
✳✲
y
=
x
▲✯ ✳✬
✬ ▲✪
✰ ✱✲✳ ✬
f
n
ê✲✳ ✬✺
✲ ✫
ù ✬✭ü
✹ ✬
f
n
(x)
≤
f
(x)
ê ✯✭✯
✮
✲✳✲
n
✬x
❸ ✬✫✮✯ê ✭
✲ù ✯✭ü
✹ ✬
lim
n
→∞
f
n
(x)
≥
f(x).
✩✬ ✁✯✮
✲ î
✫✬
f(x)
> α
✬▲✮ ✱✲✬
÷ ✪✫✮✬
✹ ✺
✳ ✪✫✼
✲
D
r
(x)
⊂
K
✳ ✬ ✭✯✪
✲
r
✬ ✼✬▲✮✭ ✲✬✺
x
î ✮✯í ü
✹ ✬î
f
(y)
> α,
ê ✯✭✯✮
✲✳✲
y
∈
D
r
(x)
✆ ✯✺
✲ ✫
ê ✭
✲ù ✯✭ ü
✹ ✬
f
n
(y)
≥
α
ý✯
✠ ✬▲
✳✲
A
=
inf
y
∈
D
r(
x
)
{
f
(y) +
n
|
x
−
y
|}
✬
B
=
inf
y
∈
D
r(
x
)\
K
{
f
(y) +
n
|
x
−
y
|}
✮✬✺
✲ ✫
f
n
(x) = min (A, B)
.
☎
✲ ✮✬ü
✹ ✬
A
≥
inf
y
∈
D
r
(
x
)
{
f(y)
}
> α
ï✞
✲ ✭✯î
ê ✯✭✯
y
∈
D
r
(x)
\
K
î✮
✲ ✺✬✺
✲ ✫
n >
(M
+
α)/r
î ✲▲
✳ ✬
−
M
❷ ✹✺
í ✪✺✪✮✬
✪▲✁✬✭✪
✲ ✭
ê ✯✭✯
f
✬✺K
î ✬✼✲ ▲✼
í✹ ì✺
✲ ✫ü
✹ ✬
f
(y)+n
|
x
−
y
|
> α
✵ ✲✭✮✯▲✮
✲ î
B > α
õ ✲▲✫✬ü
✹
✬▲✮✬✺✬▲✮✬î
f
n
(x)
≥
α
✩✬✫✮✯ ✁
✲ ✭✺✯î
ë ✯✫✮✯
✮
✲ ✺✯✭✺
✲ ✫
α
=
f(x)
−
ε
✼ ✲✺
ε >
0
ï✫✫✪✺îlim
n
→∞
f
n
(x)
≥
f(x)
−
ε
î ✁✯
✠ ✬▲
✳✲
ε
→
0
î ✼þ ✬✞✯✺
✲ ✫
✯ ✼
✲ ▲✼
í✹ ✫
✱✲ ✳ ✬✫✬ð
✯
✳ ✯
õ
✲ ▲✫✬ü
✹
✬▲✮✬✺✬▲✮✬î
lim
n
→∞
f
n
(x) =
f(x)
õ
✲ ▲✼
í✹ ✪▲
✳✲ ✯
✳ ✬✺
✲ ▲✫✮✭✯
✰ ✱✲ ✳✲ ✮✬
✲ ✭✬✺✯
■
× ÞÝ ❂④ ✽④
⑤☛
⑥⑦
f
❼ ⑨⑦ ❺❼❶ ❽ ❾
✎ ❷ ☛
⑨ ✌➁✍
✎
❶❹➈❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❼
⑨ ✍
✎ ❶
⑥ ❼❶❹
✎
X
⊆
C
✏ ❿❶❹❾
✎
⑧ ⑦
❺❼❶ ❽ ❾
✎
g(x) = lim inf
y
→
x
f(y)
❹⑦
⑨ ➤
➀ ⑨
➀ ❷
☛
⑨ ✌➁✍
✎
❶❹➈❶❼⑦✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨
X
⊆
C
✏③
× Þ Ù
ã Úä
ßÝ ❀❁Ù
✵ ✆ ✯✺
✲ ✫
✹ ✫✯✭
✲ ✪✮✬✺
✪ë✫✳✲ ✄✬
✲
✭✬✺✯❻❹ ✵✬
í✲ ✁
✯✮
✲✳ ✬
f
✫✬✭✫✬✺✪ ✷✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬î✫✬✞
✹ ✬ ü
✹ ✬
g(x)
≥
f(x)
✵ ✲✭✮✯▲✮
✲ î
lim inf
x
→
x0
g(x)
≥
lim inf
x
→
x0
f
(x) =
g(x0),
✼
✲ ▲✼
í✹ ✪▲
✳✲ ✯
✳ ✬✺
✲ ▲✫
✮ ✭
✯
✰ ✱✲ ✳✲í ✬✺
✯
✩✬✺
✲✳✲ ✫✬✺✬
íþ ✯▲✮✬î
ê✲✳ ✬✺
✲ ✫
ê ✭
✲ù ✯✭ü
✹ ✬
h(x) = lim sup
y
→
x
f(y)
î❷✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✫
✹ê ✬
✷
✭✪
✲
✭✺✬▲✮✬
➋
× Ù
ß× ÞÝ ❂④▲④
⑤☛
f
➀ ❷☛
⑨ ✌➁✍
✎
❶❹➈❶❼⑦ ✌❶❺☛
❻ ✌
✎❻⑨ ☛❶❹☛ ☛
⑨ ❼
⑨ ✍
✎ ❶
⑥ ❼❶❹
✎
❺☛✍➇⑦☞
✎
K
⑧ ☛❶❹❾✎
↔↕➙➛➜➝ ➞↕➝➟↕➠➡➠
➀ ✍
✎ ❶❹➈❶❼⑦
☛
⑨ ❼
⑨ ✍
✎ ❶
⑥ ❼❶❹
✎
G
δ
⊂
K
☞☛❶
❷✎ ☛
⑨
K
✏③
× Þ Ù
ã Úä
ßÝ ❀❁Ù
✵ ✵
✲✳ ✬✺
✲ ✫ ✫
✹ê✲ ✭ ü
✹ ✬
f
❷ í✪✺✪✮✯
✳ ✯î
✼✯✫
✲ ✼
✲ ▲✮✭ ✻✭✪
✲ ë ✯✫✮✯
✯
êí ✪✼✯✭✺
✲ ✫
✯
✁
✹ ▲
✰ ✱✲ ✯✭✼
✲
✮✯▲✞✬▲✮✬
✯
f
î ê✲✪✫ ✬
í ✯
✯✪▲
✳ ✯
✫✬✭ ✻
✹ ✺✯
✁
✹ ▲
✰ ✱✲ ✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✪▲✁✬✭✪
✲
✭✺✬▲✮✬î
✹ ✺✯
ù ✬
✠ ü
✹ ✬
✯ ✁
✹ ▲
✰ ✱✲ ✯✭✼
✲
✮✯▲✞✬▲✮✬❷ ✼
✲ ▲✮ ì▲
✹ ✯î
✫✬✞
✹ ✬ ü
✹ ✬
✯
✪✺✯✞✬✺ ✪▲
ù ✬✭✫✯
✳ ✬ ✼
✲ ▲
ð
✹ ▲✮
✲
✯
ë ✬✭✮
✲ ❷✯✪▲
✳ ✯
✹ ✺✼
✲ ▲
ð
✹ ▲✮
✲ ✯
ë ✬✭✮
✲ î
✪✫✮
✲ê✲ ✭ü
✹ ✬
{
z
: arctan(f
(z))
> α
}
=
f
−1
(
{
z
:
f
(z)
>
arctan(α)
}
)
✬ ✮✬✺✲ ✫
✹ ✺✯
ë ✪ð✬
✰ ✱✲ ✬▲✮✭✬
✲ ✫
ê✲ ▲✮
✲ ✫
✳ ✬
✳ ✬✫✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳ ✬
✳ ✬
f
✬ ✲✫
ê✲ ▲✮
✲ ✫
✳ ✬
✳ ✬✫✼
✲ ▲✮✪▲
✹ ✪
✳ ✯
✳ ✬
✳ ✬
arctan(f
)
✩✬
✬ ▲✪✺
✲ ✫
✯
✲ ✫✼✪
í ✯
✰ ✱✲✳ ✬
f
✼ ✲✺
✲
o
f
(z) := lim sup
w
→
z
f(w)
−
lim inf
w
→
z
f
(w),
✬▲✮
✱✲ê ✯✭✯
✮
✲✳✲
m
= 1,
2, . . . ,
✲ ✼✲ ▲ð
✹ ▲✮
✲
V
m
:=
{
z
∈
K
:
o
f
(z)
≥
1
m
}
❷ ✁✬✼
þ ✯
✳✲ î
ê✲ ✪✫
ê ✬
í✲ ô✬✺✯
❻❹
o
f
❷✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯
✫
✹ê ✬✭✪
✲
✭✺✬▲✮✬✬
ê ✬
í✲ ✯▲ ✻
í✲ ✞
✲✳✲ ✄✬
✲ ✭✬✺✯
❻❹
ê ✯✭✯
✁
✹ ▲
✰
✣✬✫ ✫✬✺✪
✷ ✼
✲ ▲✮ ì▲
✹ ✯✫ ✫
✹ê ✬✭✪
✲
✭✺✬▲✮✬ ✫✬✞
✹ ✬ ü
✹ ✬
V
m
❷ ✁✬✼
þ ✯
✳✲ ï✪▲
✳ ✯
V
m
▲
✱✲ ❷
✳ ✬▲✫
✲ ✬✺ ü
✹ ✯
í ü
✹ ✬✭ ✫
✹ë ✼
✲ ▲
ð
✹ ▲✮
✲ ✯
ë ✬✭✮
✲ ✭✬
í ✯✮✪
ù
✯✺✬▲✮✬
✯
K
✵✲ ✪✫ ✫✬ ✫
✹ê✲ ✭✺
✲ ✫ ü
✹ ✬ ✬
÷ ✪✫✮✯
✹ ✺
✯
ë ✬✭✮
✲
U
⊂
K
î✮✯ íü
✹ ✬î
V
m
✫✬
ð✯
✳ ✬▲✫
✲ ✬✺
K
î ✬▲✮✱✲ê✲✳ ✬✭✬✺
✲ ✫✬✫✼
✲íþ ✬✭
✹
✺✯✫✬ü
✹ ó▲✼✪✯
✳ ✬
ê✲ ▲✮
✲ ✫
x
n
∈
V
n
∩
U
ï✫✫✪✺î
o
f
(x
n
+1)
≥
1
n
+ 1
, n
≥
1,
✳ ✬
✲ ▲
✳ ✬✼
✲ ▲✼
í✹ ì✺
✲ ✫ü
✹ ✬
n
X
j
=1
o
f
(x
j
+1)
≥
n
X
j
=1
1
j
+ 1
.
ô
✲ ✞
✲ î
o
f
▲
✱✲ ❷
í ✪✺✪✮✯
✳ ✯
✬✺
K
î ✼✲
▲✮✭✯✭✪✯▲
✳✲✲ ✯▲ ✻
í✲ ✞
✲✳✲ ✄✬
✲
✭✬✺✯❻❻î
ê ✯✭✯
✁
✹ ▲
✰
✣✬✫✫✬✺✪
✷
✼
✲ ▲✮ ì▲
✹ ✯✫✫
✹ê ✬✭✪
✲
✭✺✬▲✮✬ ✵
✲ ✭✮✯▲✮
✲ î
✲ ✼
✲ ▲
ð
✹ ▲✮
✲
V
m
✮✬✺✪▲✮✬✭✪
✲ ✭
ù ✯
✠ ✪
✲ î
✬✫✬✞
✹ ▲
✳✲✲ ✄✬
✲ ✭✬✺✯
✳
✬ ò✯✪✭✬îü
✹ ✬
ê✲✳
✬ ✫✬✭ ✬▲✼
✲ ▲✮✭✯
✳✲
✬✺ ✐❹❻î
ê ✯✞ ❑
❖ ❥
î
∪
∞
m
=1
V
m
✮✯✺
ë
❷✺ ✮✬✺ ✪▲✮✬✭✪
✲ ✭
ù ✯
✠ ✪
✲
✵
✲ ✭✮✯▲✮
✲ î
G
= (
∪
∞
m
=1
V
m
)
c
❷
✹ ✺✼
✲ ▲
ð
✹ ▲✮
✲
G
δ
î
✳ ✬▲✫
✲ ✬✺
K
✲ ▲✳ ✬
f
❷✼ ✲▲✮ ì▲
✹ ✯
❑ ✄✬
✲ ✭✬✺✯
✳ ✯
❸ ✬
ê
✭✬✫✬▲✮✯
✰ ✱✲ ✳ ✬
❸ ✪✬✫
✠ ✼
✲ ✺
✲
✬✺ ✐❹❻î
ê ✯✞
❹❑ ❊ ❥
ü
✹ ✬
✳ ✪
✠ ü
✹ ✬
✮
✲✳✲ ✁
✹ ▲
✷
✼✪
✲ ▲✯
í í ✪✺✪✮✯
✳✲ ✬✺
C(K
)
î ✼✲ ✺
K
✼ ✲✺
ê ✯✼✮
✲ ✳ ✬
C
ê✲✳✬✫✬✭ ✭✬
ê
✭✬✫✬▲✮✯
✳✲ ê✲ ✭
Lf
=
Z
f dµ
ê ✯✭✯
✯
í ✞
✹
✺✯✺✬
✳ ✪
✳ ✯
✳ ✬ò
✲ ✭✬
í
µ
ï í❷✺
✳✲ ✺✯✪✫î
✯ ▲
✲ ✭✺✯
✳ ✬
L
✼ ✲✪▲✼✪
✳ ✬✼
✲ ✺
✯
✺✯✫✫✯
✮
✲ ✮✯
