Phương trình hàm cauchy và một số biến thể của nó

..

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❇Ị■ ❚❍➚ ❍➀◆●

P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨
❱⑨ ▼❐❚ ❙➮ ❇■➌◆ ❚❍➎ ❈Õ❆ ◆➶

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆✱ ✷✵✶✼


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈
✖✖✖✖✖✖✕♦✵♦✖✖✖✖✖✖✕

❇Ị■ ❚❍➚ ❍➀◆●

P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❈❆❯❈❍❨
❱⑨ ▼❐❚ ❙➮ ❇■➌◆ ❚❍➎ ❈Õ❆ ◆➶
❈❤✉②➯♥ ♥❣➔♥❤✿ P❤÷ì♥❣ ♣❤→♣ ❚♦→♥ sì ❝➜♣
▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈
◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿

❚❙✳ ◆●❯❨➍◆ ✣➐◆❍ ❇➐◆❍

❚❍⑩■ ◆●❯❨➊◆✱ ✷✵✶✼


✐

▲❮■ ❈❷▼ ❒◆
▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ồ ồ ồ
ữợ sỹ ữợ ❞➝♥ ❝õ❛ ❚❙✳◆●❯❨➍◆ ✣➐◆❍ ❇➐◆❍✳ ❚→❝ ❣✐↔ ①✐♥
tr➙♥ trå♥❣ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐
t t t ữợ ✤ë♥❣ ✈✐➯♥ ❦❤➼❝❤ ❧➺ ✈➔
t↕♦ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥
❝ù✉ ❧✉➟♥ ✈➠♥✳
◗✉❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❣û✐ ớ ỡ tợ t ổ tr
trữớ ồ ❑❤♦❛ ❤å❝ ✲ ✣↕à ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ♥â✐ ❝❤✉♥❣ ✈➔ ❝→❝ t❤➛②
❝æ tr♦♥❣ ❦❤♦❛ ❚♦→♥ ✲ ❚✐♥ ❤å❝ ♥â✐ r✐➯♥❣ ✤➣ ❞↕② ❜↔♦ ✈➔ ❞➻✉ ❞➢t t→❝ ❣✐↔
tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ q✉❛✳
❚→❝ ❣✐↔ ❝ơ♥❣ ①✐♥ ❝↔♠ ì♥ ❣✐❛ ✤➻♥❤✱ ❜↕♥ ỗ tt ồ
ữớ q t ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤
❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦
❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✻ ♥➠♠ ✷✵✶✼
❍å❝ ✈✐➯♥

❇ò✐ ❚❤à ❍➡♥❣


✐✐

▼ư❝ ❧ư❝
▼Ð ✣❺❯✳
✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳

✶
✺

✶✳✶

❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✷

P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✸

P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tê♥❣ q✉→t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✸

✶✳✹

▼ët sè ❜➔✐ t♦→♥ ù♥❣ ❞ö♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✷ ▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ù♥❣
❞ư♥❣✳
✷✺
✷✳✶

❚✐➳♣ ❝➟♥ ❣✐→ trà ❜❛♥ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✺

✷✳✶✳✶

❚r÷í♥❣ ❤đ♣ ❦❤✐ eif ❧➔ ✤ë ✤♦ ✤à❛ ♣❤÷ì♥❣✱ Rn ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✶✳✷

P❤➨♣ t➼♥❤ ❣➛♥ ✤ó♥❣ ❣✐→ trà ❜❛♥ ✤➛✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✾

✷✳✶✳✸

❚r÷í♥❣ ❤đ♣ eif ❧➔ ✤♦ ✤÷đ❝✱ ❤➻♥❤ ①✉②➳♥ ❚♦♣♦ ✳ ✳ ✳ ✳

✸✾

✷✳✷

P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② tr➯♥ ♠✐➲♥ ❤↕♥ ❝❤➳✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✷✳✸

▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✷✳✸✳✶

P❤÷ì♥❣ tr➻♥❤ ❏❡♥s❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✺

✷✳✸✳✷

P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✻

✷✳✸✳✸

P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❧✉➙♥ ♣❤✐➯♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✼

✷✳✸✳✹

P❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✽

✷✳✸✳✺

❚➼♥❤ ê♥ ✤à♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✾

▼ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✸

✷✳✹

❑➌❚ ▲❯❾◆✳
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦✳

✺✺
✺✼




é
ỵ ồ t
ởt ữỡ tr ữủ ♥❤✐➲✉ ♥❣÷í✐ ❜✐➳t ✤➳♥ ✈➔ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝ì ❜↔♥
tr♦♥❣ ỵ tt ữỡ tr ữỡ tr P❤÷ì♥❣

tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ❤❛② ✈➔ ❦❤â ❝õ❛ t♦→♥ ❤å❝ sì
❝➜♣✱ ♥â ❝â ♥❤✐➲✉ ù♥❣ ử tr ỵ tt ữỡ tr tr
ỹ t ồ ồ ỗ ❤å❝ ❣✐↔✐ t➼❝❤✱ ♥❣❤✐➯♥
❝ù✉ ❣✐↔✐ t➼❝❤✱ ❣✐↔✐ t➼❝❤ ♣❤ù❝✱ ①→❝ ①✉➜t t❤è♥❣ ❦➯✱ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ✤ë♥❣ ❧ü❝
❤å❝✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❝ì ❤å❝ ❝ê ✤✐➸♥✱ ❝ì ❤å❝ t❤è♥❣ ❦➯ t
ồ
Pữỡ tr ữủ ợ t tr♦♥❣ s→❝❤ ❝õ❛ æ♥❣ tø
♥➠♠ ✶✽✷✶✳ ❈❛✉❝❤② ✤➣ ♣❤➙♥ t➼❝❤ ❝❤➦t ❝❤➩ ♣❤÷ì♥❣ tr➻♥❤ ✤â tø ❝→❝ ❣✐↔
t❤✉②➳t r➡♥❣ ❤➔♠ sè f ❜➜t ❦➻ ❧➔ ♠ët ❤➔♠ sè ❧✐➯♥ tö❝ tø R ✤➳♥ R ✈➔ ❝→❝
❜✐➳♥ x, y ❝â t❤➸ ❧➔ ❝→❝ sè t❤ü❝ ❜➜t ❦➻✳ ●❛✉ss ❝ô♥❣ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣
tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tr♦♥❣ ❝✉è♥ s→❝❤ ❝õ❛ ỉ♥❣ tø ♥➠♠ ✶✽✵✾✱ ♥❤÷♥❣ sü ♥❣❤✐➯♥
❝ù✉ ♥➔② ❦❤ỉ♥❣ ❝❤➦t ❝❤➩ ✈➔ ụ ổ ró r r ỳ trữợ
ỳ ✶✼✾✹✱ t❛ ❝â t❤➸ t➻♠ t❤➜② tr♦♥❣ s→❝❤ ❝õ❛ ▲❡❣❡♥❞r❡✱ ð ♣❤➛♥ ❞➔♥❤
❝❤♦ sü ♥❣❤✐➯♥ ❝ù✉ t➾ sè ❞✐➺♥ t➼❝❤ ❝õ❛ ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t✱ ❝↔ ù♥❣ ❞ö♥❣ ✈➔
♣❤➙♥ t➼❝❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t✉② ♥❤✐➯♥ ❝❤ó♥❣ ✈➝♥ ❝❤÷❛ ❝❤➦t
❝❤➩ ✈➔ ❦❤æ♥❣ rã r➔♥❣✳ ❉♦ ✤â ♥â ✤➣ t❤✉ út sỹ ú ỵ ừ t tr
tớ ❣✐❛♥ ❞➔✐✳ ❑❛♥♥❛♣♣❛♥ ✤➣ ✈✐➳t✿ ✏❈→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✤➣ ✤❛♠
♠➯ ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❬P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ✸ ❦✐➸✉ t÷ì♥❣
✤÷ì♥❣❪✱ ✈➔ sü ↔♦ t÷ð♥❣ ♥➔② s➩ t✐➳♣ tử t q tú
ỡ
ữợ ❝❤✉♥❣ ❝õ❛ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧➔ sû
❞ư♥❣ ♥❤✐➲✉ ❧♦↕✐ ✤✐➲✉ ❦✐➺♥ t❤ỉ♥❣ t❤÷í♥❣ tr➯♥ ❤➔♠ sè ❜➜t ❦➻✳ ◆â ❝❤➾ r❛


✷

r➡♥❣ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❦❤✐ f : R R ộ s
r sỹ tỗ t ❝õ❛ c ∈ R✱ s❛♦ ❝❤♦ f (x) = cx✱ ợ ồ x R tỹ t
ữủ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♥❤✐➲✉ ❝→❝❤✳ ❱➼ ❞ö✱ ❈❛✉❝❤② ✤➣ ❣✐↔ sû f ❧✐➯♥
tö❝✳ ❉❛r❜♦✉① ✤➣ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ f ❝â t❤➸ ✤÷đ❝ ❣✐↔ t❤✐➳t ❤♦➦❝ ✤ì♥ ✤✐➺✉

❤♦➦❝ ❜à ❝❤➦♥ tr➯♥ ♠ët ❦❤♦↔♥❣✱ ❋r➨❝❤❡t✱ ❇❧✉♠❜❡r❣✱ ❇❛♥❛❝❤✱ ❙✐❡r♣✐♥s❦✐✱
❑❛❝✱ ❆❧❡①✐❡✇✐❝③✲❖r❧✐❝③✱ ✈➔ ❋✐❣✐❡❧ ✤➣ ❣✐↔ t❤✐➳t r➡♥❣ f ❧➔ ✤♦ ✤÷đ❝ ▲❡❜❡s❣✉❡✱
❑♦r♠❡s ✤➣ ❣✐↔ t❤✐➳t r➡♥❣ f ❜à ❝❤➦♥ tr➯♥ t➟♣ ✤♦ ✤÷đ❝ ❞÷ì♥❣✱ ❖str♦✇s❦✐
✈➔ ❑❡st❡❧♠❛♥ ✤➣ ❣✐↔ t❤✐➳t r➡♥❣ f ❜à ❝❤➦♥ tø ♠ët ❜➯♥ tr➯♥ t➟♣ ✤♦ ✤÷đ❝
❞÷ì♥❣✱ ✈➔ ▼❡❤❞✐ ✤➣ ❣✐↔ t❤✐➳t r➡♥❣ f ❜à ❝❤➦♥ tr➯♥ tr➯♥ t➟♣ ♥❤â♠ ❇❛✐r❡✳
▼➦t ❦❤→❝✱ ❍❛♠❡❧ ✤➣ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❦❤✐ ❦❤æ♥❣ ❝â
❜➜t ❦➻ ✤✐➲✉ ❦✐➺♥ ❦❤→❝ ❝õ❛ f ✳ ❇➡♥❣ ✈✐➺❝ sû ❞ư♥❣ ❝ì sð ❍❛♠❡❧✱ ỉ♥❣ t❛ ✤➣ s✉②
r❛ r➡♥❣ ❝â ♥❤✐➲✉ ♥❣❤✐➺♠ ❦❤ỉ♥❣ t✉②➳♥ t➼♥❤ tø ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②
✈➔ ỉ♥❣ ✤➣ t➻♠ r❛ t➜t ❝↔ ❝❤ó♥❣✳
P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✤➣ ✤÷đ❝ ❦❤→✐ q✉→t ❤â❛ ❤❛② ❜ê s✉♥❣ t
ữợ ởt ữợ ✤à♥❤ ✈➔ ♠✐➲♥ ❣✐→ trà
❝õ❛ f t❤➔♥❤ ❝→❝ ♥❤â♠ ❝õ❛ ❧♦↕✐ ♥➔♦ ✤â✱ ✈➼ ❞ư ✭❝♦♠♣❛❝t ✤à❛ ♣❤÷ì♥❣✮ ♥❤â♠
P♦❧✐s❤✱ ✈➔ ✤➸ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ f t❤ä❛ ♠➣♥ ♠ët ✤✐➲✉ ❦✐➺♥ ❜➜t ❦➻ ❝õ❛
❣✐↔ t❤✉②➳t ✤♦ ✤÷đ❝ ✭❇❛✐r❡✱ ❍❛❛r✱ ❤❛② ❈❤r✐st❡♥s❡♥✮✱ ✈➔ ❝â t❤➸ ❧➔ ❝→❝ ❣✐↔
t❤✉②➳t ❝ë♥❣ t➼♥❤✱ t❤➻ õ tử ú ỵ t õ ữỡ tr➻♥❤
❤➔♠ ❈❛✉❝❤②✱ t❛ ♥â✐ ❧➔ ❤➔♠ t❤✉➛♥ ♥❤➜t ❤♦➦❝ ❝ë♥❣ t ởt ữợ
ủ ử tt tứ t ữủ ữ ữợ tờ qt
t ✤ê✐ ✤à♥❤ ♥❣❤➽❛ ♠✐➲♥ ①→❝ ✤à♥❤ ❝õ❛ f ✤➸ ♠➔ s➩ ❦❤ỉ♥❣ ❝â ♠ët ❝➜✉
tró❝ ✤↕✐ sè ✤➭♣ ♥ú❛✱ ♠➔ ❧➔ s➩ ❝❤➾ ✤ì♥ t❤✉➛♥ ❧➔ t➟♣ ❝♦♥ ♥➔♦ ✤â ừ
tự ử ởt t ỗ ♣❤➛♥ ❜ị ❝õ❛ t➟♣ ✤♦ ✤÷đ❝ ✵✱ ✈✈✳
❙ü ❜✐➳♥ ✤ê✐ ♥➔② ❧➔ ✤➸ t❤❛② ✤ê✐ ♠✐➲♥ ❣✐→ trà ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ✈➼ ❞ư✱ ❣✐↔
sû r➡♥❣ f t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❝❤➾ ✈ỵ✐ ❝➦♣ (x, y) t❤✉ë❝
✈➔♦ t➟♣ ❝♦♥ ❝õ❛ R2n ✱ ✈➼ ❞ö ✤❛ t↕♣ ✭✈➔ f ❝â t❤➸ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t♦➔♥
❦❤ỉ♥❣ ❣✐❛♥ ❤♦➦❝ tr➯♥ t➟♣ ❝♦♥ ❝õ❛ ♥â✮✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ ♥➔②✱
t❛ õ t t sỹ tỗ t ổ t t➼♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤
❤➔♠ ❈❛✉❝❤② ✭♠➦❝ ❞ị ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➲✉ ♠↕♥❤✮ ❤♦➦❝ ✭tr♦♥❣ tr÷í♥❣ ❤đ♣
❦❤✐ f ✤÷đ❝ ❣✐↔ ✤à♥❤ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥✮ f ♣❤↔✐ t❤ä❛ ♠➣♥
♥â ✈ỵ✐ t➜t ❝↔ ❝→❝ ❝➦♣ (x, y) ❝â t❤➸✳
❇➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â ❧➔ ❝æ♥❣

✸

❝ư ✤➸ ❣✐↔✐ q✉②➳t r➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❤❛② ✈➔ ❦❤â✱ ♥â ①✉➜t
❤✐➺♥ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ✤➲ t ồ s ọ tr ữợ qố t tữớ
ởt t tự ố ợ ồ s t ❧✐➺✉ ✈➔ ❝→❝ ✤➲ t➔✐ ✈➲ ♣❤÷ì♥❣
tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✤➣ ✤÷đ❝ ❜✐➯♥ s♦↕♥ ✈➔ t❤ü❝ ❤✐➺♥✳ ❚✉② ♥❤✐➯♥ ♠é✐ t➔✐ ❧✐➺✉
❝❤➾ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ✈➔ ❝→❝ ù♥❣ ❞ư♥❣✱ ❝❤÷❛ ❜❛♦ q✉→t ✤÷đ❝ ✤➛② ✤õ✳
❱➻ ✈➟②✱ ❝→❝ ✈➜♥ ✤➲ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➝♥ ❝á♥ r➜t ♣❤♦♥❣ ♣❤ó✳

✷✳ ▼ư❝ ✤➼❝❤✳
▼ư❝ ✤➼❝❤ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ✈➜♥ ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣
tr➻♥❤ ❤➔♠✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â✱ ①❡♠ ①➨t
❦❤↔ ♥➠♥❣ ❣✐↔✐ ✤÷đ❝ ✈➔ sü ê♥ ✤à♥❤ t÷ì♥❣ ✤è✐ ❝õ❛ ♥â ✤è✐ ✈ỵ✐ ❝→❝ t➟♣ ❝♦♥
❝õ❛ ❦❤ỉ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ởt số ợ ữủ tr
❝❤➥♥❣ ❤↕♥ ♥❤÷ ♠ët ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✤â ♠ët sè ♠ơ ♣❤ù❝ t↕♣ ❝→❝
❤➔♠ ❝❤÷❛ ❜✐➳t✳✳✳✳ ❈→❝ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ♠ð rë♥❣ ✤➳♥ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ t ự ử ỵ tt tr
ỗ ữù tự t ồ ❝❤♦ ❤å❝ s✐♥❤ ❚❍P❚ ✈➔ ❧➔
t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❝❤♦ s✐♥❤ ✈✐➯♥ ♥❣➔♥❤ ❚♦→♥ ❤å❝✳

✸✳ ✣è✐ t÷đ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉✳
✣è✐ t÷đ♥❣ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ ❧✉➟♥ ✈➠♥ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔
♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â✳ ▼ët ❝→❝❤ ❝ö t❤➸✱ ❧✉➟♥ ✈➠♥ s➩ tr➻♥❤ ❜➔② ❝→❝ ❦➳t
q✉↔ ❝❤➼♥❤ tr♦♥❣ ❝→❝ t➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪✱ ❬✷❪✱ ❬✸❪ ✈➔ ❝→❝ ❜➔✐ ❜→♦ ❬✹❪✱ ❬✺❪✳

✹✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉✳
❚❤✉ t❤➟♣ ❝→❝ ❜➔✐ ❜→♦ ❦❤❛♦ ❤å❝ ✈➔ t➔✐ ❧✐➺✉ ❝õ❛ ❝→❝ t→❝ ❣✐↔ ♥❣❤✐➯♥ ❝ù✉
❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ự ử
r ờ q ợ t ữợ ❞➝♥ ✈➲ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣

tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â✳


✺✳ ❇è ❝ö❝ ❧✉➟♥ ✈➠♥✳

✹

❚→❝ ❣✐↔ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉ ở tữỡ ự ợ
ữỡ

ữỡ Pữỡ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳
✶✳✶✳ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳
✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳
✶✳✸✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tê♥❣ q✉→t✳
✶✳✹✳ ▼ët sè ❜➔✐ t♦→♥ ù♥❣ ❞ư♥❣✳
❈❤÷ì♥❣ ✷✳ ▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔
ù♥❣ ❞ư♥❣✳
✷✳✶✳ ❚✐➳♣ ❝➟♥ ❣✐→ trà ❜❛♥ ✤➛✉✳
✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② tr➯♥ ♠✐➲♥ ❤↕♥ ❝❤➳✳
✷✳✸✳ ▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤②✳
✷✳✹✳ ▼ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛✳


✺

❈❤÷ì♥❣ ✶

P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔②✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ t➼♥❤ ❝❤➜t ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ❤➔♠✳ ❚r♦♥❣ ✤â✱ t→❝ ❣✐↔ ✤✐ s➙✉ ✈➲ ♥❣❤✐➯♥ ❝ù✉ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠

❈❛✉❝❤② ✈➔ ♠ët sè ❜➔✐ t♦→♥ ù♥❣ ❞ư♥❣✳
◆ë✐ ❞✉♥❣ ❝❤➼♥❤ ✤÷đ❝ t❤❛♠ ❦❤↔♦ t↕✐ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✷❪✱ ❬✸❪✳

✶✳✶ ❚ê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳
✣à♥❤ ♥❣❤➽❛ ✶✳✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ♠➔ ➞♥ ❧➔ ❝→❝ ❤➔♠
sè✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tù❝ ❧➔ t➻♠ ❝→❝ ❤➔♠ sè ❝❤÷❛ ❜✐➳t ✤â✳

❚✐➳♣ ❝➟♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ♠é✐ ♥❣÷í✐ ❝â ♥❤ú♥❣ ❝ì sð ✈➔ ♣❤÷ì♥❣ ♣❤→♣
❦❤→❝ ♥❤❛✉✳ ❚✉② ♥❤✐➯♥✱ ❞ü❛ ✈➔♦ ✤➦❝ tr÷♥❣ ❝õ❛ ❝→❝ ❤➔♠ t❛ ❝â t❤➸ ①➙② ❞ü♥❣
✤÷đ❝ ♠ët số ữợ ữ s
tr ♣❤ị ❤đ♣✿ ❍➛✉ ❤➳t ❝→❝ ❣✐→ trà ❜❛♥ ✤➛✉ ❝â t❤➸ t❤➳
✈➔♦ ❧➔✿ x = 0, x = 1, ...❀ tø ✤â t➻♠ r❛ ♠ët t➼♥❤ ❝❤➜t q✉❛♥ trå♥❣ ♥➔♦
✤â ❤♦➦❝ ❝→❝ ❣✐→ trà ✤➦❝ ❜✐➺t ❝õ❛ ❤➔♠ ❤♦➦❝ t➻♠ ❝→❝❤ ❝❤ù♥❣ ♠✐♥❤ ❤➔♠
sè ❤➡♥❣✳
✷✳ ◗✉② ♥↕♣ t♦→♥ ❤å❝✿ ✣➙② ❧➔ ♣❤÷ì♥❣ ♣❤→♣ sû ❞ư♥❣ ❣✐→ trà f (x) ✈➔ ❜➡♥❣
❝→❝❤ q✉② ♥↕♣ ✈ỵ✐ n ∈ N ✤➸ t➻♠ f (n)✳ ❙❛✉ ✤â t➻♠ f ( n1 ) ✈➔ f (e)✳ P❤÷ì♥❣
♣❤→♣ ♥➔② t❤÷í♥❣ →♣ ❞ư♥❣ tr♦♥❣ ❜➔✐ t♦→♥ ♠➔ ð ✤â ❤➔♠ f ✤➣ ✤÷đ❝ ①→❝
✤à♥❤ tr➯♥ Q❀ tø ✤â ♠ð rë♥❣ tr➯♥ ❝→❝ t➟♣ sè rë♥❣ ❤ì♥✳
✸✳ ❙û ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ✈➔ ❦✐➸✉ ❈❛✉❝❤②✳


✻

✹✳ ◆❣❤✐➯♥ ❝ù✉ t➼♥❤ ✤ì♥ ✤✐➺✉ ✈➔ t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ ❝→❝ ❤➔♠✳ ❈→❝ t➼♥❤ ❝❤➜t
♥➔② →♣ ❞ư♥❣ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❤♦➦❝ ❦✐➸✉ ❈❛✉❝❤②✳ ❈→❝
♣❤÷ì♥❣ tr➻♥❤ ✤â ♥➳✉ ❦❤ỉ♥❣ ❝â t➼♥❤ ✤ì♥ ✤✐➺✉✱ ❧✐➯♥ tư❝ t❤➻ ❜➔✐ t♦→♥ trð
♥➯♥ ♣❤ù❝ t↕♣ ❤ì♥ ♥❤✐➲✉✳
✺✳ ❚➻♠ ✤✐➸♠ ❝è ✤à♥❤ ❤♦➦❝ ❣✐→ trà ✵ ❝õ❛ ❝→❝ ❤➔♠✳
✻✳ ◆❣❤✐➯♥ ❝ù✉ t➼♥❤ ✤ì♥ →♥❤ ✈➔ t♦➔♥ →♥❤ ❝õ❛ ❝→❝ ❤➔♠ ❧ơ② t❤ø❛ tr♦♥❣
♣❤÷ì♥❣ tr➻♥❤✳

✼✳ ❉ü ✤♦→♥ ❤➔♠ ✈➔ ❞ị♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♣❤↔♥ ❝❤ù♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉
❞ü ✤♦→♥ ✤ó♥❣✳
✽✳ ❚↕♦ ♥➯♥ ❝→❝ ❤➺ tự tr ỗ
t t t ừ số
ứ ởt số ữợ tr t t➙♠ ✤➢❝ ♣❤➛♥ ♣❤÷ì♥❣ tr➻♥❤
❤➔♠ ❈❛✉❝❤② ♥➯♥ ✤➣ ✤✐ s➙✉ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ♥â✳

✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳
✣à♥❤ ♥❣❤➽❛ ✶✳✷ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝â ❞↕♥❣✿
f (x + y) = f (x) + f (y), ∀x, y ∈ R,
tr♦♥❣ ✤â✱

f (x)

❧➔ ❤➔♠ ①→❝ ✤à♥❤ tr➯♥

✭✶✳✶✮

R✳

❍➔♠ ❢ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤

f (x + y) = f (x) + f (y), ∀x, y ∈ R,
✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ở t

ỵ số tử f (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❦❤✐
✈➔ ❝❤➾ ❦❤✐✿

f (x) = ax, x R,
tr õ

a

số tũ ỵ


✼
❈❤ù♥❣ ♠✐♥❤✿

❈❤♦ x = y ✱ t❛ ❝â✿ f (2x) = 2f (x)✳
❇➡♥❣ ❝→❝❤ q✉② ♥↕♣ t❤❡♦ n✱ t❛ s➩ ❝❤ù♥❣ ♠✐♥❤✿

f (nx) = nf (x), ∀n ∈ N, x ∈ R.
❚❤➟t ✈➟②✱
❱ỵ✐ n = 1 ✈➔ n = 2 ❤➺ t❤ù❝ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ ❧➔ ✤ó♥❣✳
●✐↔ sû✿ f (kx) = kf (x), k ≥ 1✳ ❑❤✐ ✤â✿

f ((k + 1)x) = f (x + kx) = f (x) + f (kx) = f (x) + kf (x) = (k + 1)f (x).
❈❤♦ x = y = 0✱ s✉② r❛✿ f (0) = 0✳
❚✐➳♣ t❤❡♦✱ t❤❛② x = −y ✱ t❛ ✤÷đ❝✿

0 = f (x − x) = f (x) + f (−x),
❤❛②

f (−x) = −f (x).
◆➳✉ n < 0✱ t❤➻✿ f (nx) = f ((−n).(−x)) = −nf (−x) = nf (x)✳
❱➟② f (nx) = nf (x), ∀n ∈ Z✳
◆➳✉ n ∈ Z, n = 0 t❤➻✿

f (x) = f n.

❤❛②

f

x
x
= n.f
,
n
n

x
f (x)
=
.
n
n

m
∈ Q✱ tr♦♥❣ ✤â m, n ∈ Z✳ ❚❛ ❝â✿
n
x
x
m
m
.x = f m
=mf
=
f (x) = p f (x).
f (p x) = f

n
n
n
n
[n α]
❱ỵ✐ ♠å✐ α ∈ R, n ∈ N✱ ✤➦t rn =
∈ Q✱
n
1
❚❛ ❝â rn ≤ α < rn + ✳
n
❉♦ ✤â t❛ ❝â ✿
❳➨t p =

lim rn = α

n→∞

✈➔

α x = lim (rn x).
n→∞


✽

❱➻ f ❧✐➯♥ tö❝ ♥➯♥✿

f (αx) = lim f (rn x) = lim rn f (x) = αf (x).
n→∞

n→∞

✭✶✳✷✮

✣➦t a = f (1) t❤➻ f (x) = f (x.1) = xf (1) = ax✳
❱➟② f (x) = ax, ∀x ∈ R, a R trữợ
ữủ f (x) = ax, ∀x ∈ R, a ∈ R t❤➻ ❞➵ t❤➜② r➡♥❣ f ❧➔ ♠ët
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤②✳

✣✐➲✉ ❦✐➺♥ tử ừ f tữỡ ữỡ ợ ởt tr ❝→❝ ✤✐➲✉ ❦✐➺♥
s❛✉✿

❇ê ✤➲ ✶✳✶ ❈❤♦ f : R → R ởt ừ ữỡ tr
ổ ỗ ♥❤➜t ❜➡♥❣ ✵✳ ❑❤✐ ✤â✱ ❝→❝ ♠➺♥❤ ✤➲ s❛✉ ❧➔ t÷ì♥❣ ✤÷ì♥❣✿
✭✶✮ ❢ ❧✐➯♥ tư❝ tr➯♥

R✱

✭✷✮ ❢ ❧✐➯♥ tư❝ t↕✐ ✤✐➸♠

x0 ∈ R✱

✭✸✮ ❢ ❧✐➯♥ tö❝ t↕✐ ✤✐➸♠ ✵✱
✭✹✮ ❢ ✤ì♥ ✤✐➺✉ t❤ü❝ sü tr➯♥ ♠ët ❦❤♦↔♥❣ tr♦♥❣

R✱

✭✺✮ ❢ ❜à ❝❤➦♥ tr➯♥ ♠ët ❦❤♦↔♥❣ ✭❤♦➦❝ ♠ët ✤♦↕♥✮ tr♦♥❣

R✳

❈❤ù♥❣ ♠✐♥❤✿

❚ø ♣❤➨♣ ự ừ ỵ t õ

f (rx) = rf (x), ∀x ∈ R, r ∈ Q.
❚ø (1) s✉② r❛ (2) ✈➔ tø (2) s✉② r❛ (3) ❧➔ ❤✐➸♥ ♥❤✐➯♥✳
❚❛ ❝❤ù♥❣ ♠✐♥❤✿ (3) s✉② r❛ (1)✳
❚❤➟t ✈➟②✱ ∀x0 ∈ R✱

f (x) = f (x − x0 + x0 ) = f (x − x0 ) + f (x0 ).
◆➳✉ limn→∞ (xn ) = x0 t❤➻✿

lim f (xn ) = lim [f (xn − x0 ) + f (x0 )] = f (0) + f (x0 ) = f (x0 ).

n→∞

n→∞

❱➟② f ❧✐➯♥ tö❝ t↕✐ x0 ✳


✾

❉♦ ✤â✱ f ❧➔ ❤➔♠ sè ❧✐➯♥ tö❝✳
❈❤ù♥❣ ♠✐♥❤✿ tø s r
sỷ f tử
ỵ ✶✳✶✿ f (x) = ax, ∀x ∈ R, a ❧➔ số
f ổ ỗ t a = 0✳

❉♦ ✤â f ✤ì♥ ✤✐➺✉ tr➯♥ R✳
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû f ✤ì♥ ✤✐➺✉ tr➯♥ ♠ët ❦❤♦↔♥❣ I ⊂ R✳ ❚❛ ❣✐↔ t❤✐➳t✱ f
✤ì♥ ✤✐➺✉ t➠♥❣✳
▲➜② x0 ∈ I ✈➔ ε > 0✱ s❛♦ ❝❤♦✿ (x0 − ε, x0 + ε) ⊂ I ✳
ε
ε
❱➻ x0 − < x0 < x0 + ♥➯♥✿
n
n
ε
ε
f (ε)
f (ε)
= f x0 −
< f (x0 ) < f x0 +
= f (x0 ) +
f (x0 ) −
n
n
n
n
❉♦ ✤â✿

lim f x0 −

n→∞

ε
ε
= lim f x0 +

= f (x0 )
n→∞
n
n

✭✶✳✸✮

●✐↔ sû✿ limn→∞ xn = x0 ✳
❱ỵ✐ ♠å✐ δ > 0✱ ❝❤å♥ n0 ∈ N s❛♦ ❝❤♦✿
ε
ε
< xn < x0 + , ∀n > n0 ✱
x0 −
n0
n0
ε
ε
✈➔ f x0 +
− f x0 −
< δ.
n
n
ε
ε
< f (xn ) < f x0 +
✳
❑❤✐ ✤â✱ f x0 −
n0
n0
❙✉② r❛✱ |f (xn ) − f (x0 )| < δ, ∀n > n0 ✳

❱➟② f ❧✐➯♥ tö❝ t↕✐ x0 ♥➯♥ f ❧✐➯♥ tö❝✳
❈❤ù♥❣ ♠✐♥❤✿ (1) ⇔ (5)✳ ❚❛ ❝❤➾ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤ (5) s✉② r❛ (1)✳
●✐↔ sû f ❜à ❝❤➦♥ tr➯♥ ❦❤♦↔♥❣ I ⊂ R✳
▲➜② x0 ∈ I, ε > 0✱ s❛♦ ❝❤♦✿ (x0 − ε, x0 + ε) ⊂ I ✳
❱ỵ✐ ♠å✐ x ∈ (−ε, ε) : f (x) = f (x + x0 ) − f (x0 ) ✈ỵ✐ x + x0 ∈ I ✳
❉♦ ✤â✱ f ❜à ❝❤➦♥ tr➯♥ (−ε, ε)✳ tỗ t M > 0 s |f (x)| ≤ M ✱
✈ỵ✐ |x| < ε✳
●✐↔ sû✱

lim xn = 0.

n→∞

✣➦t kn =

1
|xn |

∈ N✳ ❚❛ ❝â✿ |kn xn | ≤

1
|xn |

|xn | =

|xn |✳


✶✵

❱ỵ✐ ♠å✐ δ > 0✱ ❝❤å♥ n0 ∈ N s❛♦ ❝❤♦✿

M
< δ ✈➔ |kn xn | <
kn
❑❤✐ ✤â✱

|f (xn )| = |f (

|xn | < ε, ∀n > n0 ✳

kn xn
f (kn xn )
M
)| =
≤
< δ.
kn
kn
kn

❱➟② limn→∞ f (xn ) = 0 = f (0) ♥➯♥ f ❧✐➯♥ tö❝ t↕✐
õ f tử

ứ ỵ ờ ✤➲ ✶✳✶ ✱ s✉② r❛ ❤➔♠ f t❤ä❛ ♠➣♥ ♠ët tr♦♥❣ ❝→❝
✤✐➲✉ ❦✐➺♥ ❝õ❛ ❇ê ✤➲ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❦❤✐ ✈➔ ❝❤➾
❦❤✐✿

f (x) = ax, ∀x ∈ R,
✈ỵ✐ a ❧➔ ❤➡♥❣ sè✳

❚❛ ❝â ♠ët sè ❤➺ q✉↔ s❛✉✿

❍➺ q✉↔ ✶✳✶ ❍➔♠ sè ❢ ❧✐➯♥ tö❝ tr➯♥ R ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠
f (x + y) = f (x).f (y), ∀x, y ∈ R,
❦❤✐ ✈➔ ❝❤➾ ❦❤✐

f (x) = bx , ∀x ∈ R, b > 0

✭✶✳✹✮

❧➔ ❤➡♥❣ sè✳

❈❤ù♥❣ ♠✐♥❤✳

◆➳✉ ❝â x0 ∈ R✱ s❛♦ ❝❤♦✿ f (x0 ) = 0✱ t❤➻✿

f (x) = f (x − x0 + x0 ) = f (x − x0 )f (x0 ) = 0, ∀x ∈ R.
❚ù❝ ❧➔✱ f ≡ 0✳
❚❛ ❣✐↔ t❤✐➳t f (x) = 0, ∀x ∈ R f (x) ổ ỗ t
õ

x
x x
+
=f
2 2
2
tự tữỡ ữỡ ợ

2

f (x) = f

> 0.

ln[f (x + y)] = ln[f (x)f (y)] = lnf (x) + lnf (y),
❤❛②

g(x + y) = g(x) + g(y),




tr õ g(x) = ln[f (x)]
ỵ t ❝â✿ g(x) = ax, a ∈ R✳ ❱➟②

f (x) = eax = bx , b = ea > 0.

❍➺ q✉↔ ✶✳✷ ❍➔♠ sè ❢ ❧✐➯♥ tö❝ tr➯♥ R

{0}

❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤

❤➔♠

f (xy) = f (x) + f (y), ∀x, y ∈ R
❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿

f (x) = a ln|x|, ∀x ∈ R

{0} ,

{0}✳

❈❤ù♥❣ ♠✐♥❤✳

+) ❱ỵ✐ x, y ∈ R+ ✱ ✤➦t x = eu ✈➔ y = ev ✳
❚❛ ❝â✿

f (eu+v ) = f (eu ) + f (ev )
⇔ g(u + v) = g(u) + g(v), ∀u, v ∈ R.
tr♦♥❣ ✤â✱ g(u) = f (eu ) ❧✐➯♥ tö❝ tr R
ử ỵ t õ g(u) = a u✳
❙✉② r❛✿ f (x) = g(lnx) = a lnx, ∀x ∈ R+ ✳

+) ❱ỵ✐ x < 0✱ t❛ ❝â✿
f x2 = f (x) + f (x),
❤❛②

1
1
f x2 =
a lnx2 = a ln(−x).
2
2
❱➟② f (x) = a ln|x|, ∀x ∈ R {0} ợ a R tũ ỵ
f (x) =

q✉↔ ✶✳✸ ❍➔♠ sè ❢ ❧✐➯♥ tö❝ tr➯♥ R ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠

f (xy) = f (x)f (y), ∀x, y ∈ R
❦❤✐ ✈➔ ❝❤➾ ❦❤✐

f (x) = |x|α , ∀x ∈ R

❈❤ù♥❣ ♠✐♥❤✳

❚❤❛② y = 1✱ t❛ ✤÷đ❝✿

{0} , α

{0} ,

❧➔ ❤➡♥❣ sè✳


✶✷

f (x) = f (x).f (1)
⇔ f (x).(1 − f (1)) = 0, ∀x ∈ R.
◆➳✉ f (1) = 1 t❤➻ f (x) = 0, ∀x ∈ R

{0}✳

❉♦ ✈➟②✱ f ≡ 0✳
❳➨t f (1) = 1✳

1
1
= f (x).f

, ∀x ∈ R
x
x
❙✉② r❛ f (x) = 0, ∀x ∈ R {0}✳ ❉♦ ✤â✿

❑❤✐ ✤â✱ f (1) = f

x.

{0}✳

f x2 = f (x).f (x) = f 2 (x) > 0, ∀x ∈ R

{0} .

❙✉② r❛✱ f (x) > 0, ∀x ∈ R+ ✳
✣➦t g(t) = f (et ), t ∈ R✳ ❑❤✐ ✤â✱

g(t + u) = f (et+u ) = f (et .eu ) = f (et )f (eu ) = g(t).g(u).
P❤÷ì♥❣ tr➻♥❤ tr➯♥ ❝â ♥❣❤✐➺♠ ❧➔✿ g(t) = at , ∀t ∈ R✳
❱➻ x = et ✱ t÷ì♥❣ ✤÷ì♥❣ t = ln x✱ ♥➯♥✿

f (x) = g(ln x) = aln

x

= eln

a ln x

= eln

x ln a

= xln

a

= xα , α ∈ R.

❳➨t x, y ∈ R− ✱ ❦❤✐ ✤â −x, −y ∈ R+ .
◆➳✉ x = y ✱ t❛ ♥❤➟♥ ✤÷đ❝✿ f x2 = f 2 (x) > 0✳
❱➻ x2 > 0✱ t❤❡♦ ❝❤ù♥❣ ♠✐♥❤ tr➯♥ f x2 = x2
❉♦ ✤â✿

f 2 (x) = x2α .
❙✉② r❛ f (x) = ±|x|α ✳
❱➟② ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔✿
✭✶✮ f (x) = 0, ∀x ∈ R

{0}✳

✭✷✮ f (x) = |x|α , ∀x ∈ R
✭✸✮ f (x) =

{0}✳

|x|α , x > 0,
−|x|α , x < 0.

α

, α ∈ R✳


✶✸

✶✳✸ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② tê♥❣ q✉→t
❈❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣✿

f (x) + f (y) − f (x + y) = g(H(x, y))

✭✶✳✺✮

tr♦♥❣ ✤â✱ f ✈➔ g ❧➔ ❝→❝ ❤➔♠ ♣❤↔✐ t➻♠✱ H ❧➔ ❤➔♠ ✤➣ ❝❤♦✳ ❑❤✐ g ≡ 0 t❤➻
✭✶✳✺✮ trð t❤➔♥❤ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❈→❝ ❤➔♠ sè ð ✤➙② ✤÷đ❝ ①➨t ❧➔
❤➔♠ sè t❤ü❝✱ tù❝ ❧➔ t➟♣ ①→❝ ✤à♥❤ ✈➔ t➟♣ ❣✐→ trà ❝õ❛ ♥â ❧➔ R ❤♦➦❝ t➟♣ ❝♦♥
❝õ❛ R✳
❙❛✉ ✤➙② t❛ ①➨t ♠ët sè tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ✭✶✳✺✮✳
✰ H(x, y) = xy ✱ ❦❤✐ ✤â ✭✶✳✺✮ trð t❤➔♥❤✿

f (x) + f (y) − f (x + y) = g(xy)
✰ H(x, y) =

1 1
+ ✈➔ g = −f ✱ ❦❤✐ ✤â ✭✶✳✺✮ trð t❤➔♥❤✿
x y
f (x) + f (y) − f (x + y) = −f (x−1 + y −1 )

✰ H(x, y) =

✭✶✳✻✮

✭✶✳✼✮

xy(x + y)
✈➔ g = f ✿ ✭✶✳✺✮ trð t❤➔♥❤✿
x2 y 2 + xy
f (x) + f (y) − f (x + y) = f

xy(x + y)
+ y 2 + xy

x2

✭✶✳✽✮

❚❛ ♥❤➟♥ t❤➜② r➡♥❣✱ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮✲✭✶✳✽✮ ❧➔ ♥❤ú♥❣ t
q tở tr ỵ tt ữỡ tr

①➨t✿
✰ ◆➳✉ g(x) = c ✭g ❧➔ ❤➔♠ ❤➡♥❣✮ t❤➻ ợ t ự H
tữỡ ữỡ ợ

f (x) + f (y) f (x + y) = c.
❘ã r➔♥❣ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ tr➯♥ ❧➔✿

f (x) = A(x) + c,

✶✹

tr♦♥❣ ✤â✱ A(x) ❧➔ ♠ët ❤➔♠ ❝ë♥❣ t➼♥❤ ❜➜t ❦ý✱ ❤❛② A(x) ❧➔ ♥❣❤✐➺♠ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ✭✶✳✶✮✳
❱➟② ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔② ❧➔✿

f (x) = A(x) + c,
✈➔

g(x) = c.
✰ ❚÷ì♥❣ tü tr♦♥❣ tr÷í♥❣ ❤đ♣ H(x, y) = c✱ ❝æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ✭✶✳✺✮
❧➔✿

f (x) = A(x) + g(c) ✈➔ g ❧➔ ❤➔♠ ❜➜t ❦ý✳
tr♦♥❣ ✤â✱ A(x) ở t tũ ỵ
ú t ồ (f, g) ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ❧➔ t➛♠ t❤÷í♥❣ ♥➳✉

f ❧➔ ❛❢✐♥✱ tù❝ ❧➔ f (x) = A(x) + c✱ tr♦♥❣ ✤â A(x) ❧➔ ❝ë♥❣ t➼♥❤ ✈➔ c ❧➔
❤➡♥❣ sè✳
✰ ✣➸ þ r➡♥❣ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮✲✭✶✳✽✮ ❧➔ ❞↕♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ✈ỵ✐

H(x, y) ❝â ❞↕♥❣ s❛✉✿
H(x, y) = ψ(φ(x) + φ(y) − φ(x + y))

✭✶✳✾✮

❱➻ ❞➵ t❤➜② ✭✶✳✻✮✲✭✶✳✽✮ t÷ì♥❣ ù♥❣ ✈ỵ✐ ✈✐➺❝ ❝❤å♥✿ φ(x) = x2 , ln x, x−1
−u −u −1
, e , u ✳
✈➔ ψ(u) =

2
❇➙② ❣✐í t❛ ①➨t ♠ët tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❝õ❛ ✭✶✳✺✮ ❝â ❞↕♥❣ ♥❤÷ s❛✉

f (x) + f (y) − f (x + y) = g(φ(x) + φ(y) − φ(x + y))

✭✶✳✶✵✮

❘ã r➔♥❣✱ t ợ ồ g ữỡ tr ✭✶✳✶✵✮ ✤➲✉ ❝â ♥❣❤✐➺♠

f ❧➔ ❛❢✐♥✱ tù❝ ❧➔ ✭✶✳✶✵✮ ❝â ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣✳
❱➻ ✈➟② ❝❤ó♥❣ t❛ ①➨t φ ❦❤ỉ♥❣ ❛❢✐♥ ỵ I (, +)

[, +) (, +∞)✱ (−∞, −α]✱ (−∞, −α)✮✱ tr♦♥❣ ✤â α ≥ 0✳

✶✳✹ ởt số t ự ử
ứ ỡ s ỵ tt ✤➣ ♥➯✉ ð ♣❤➛♥ tr➯♥✱ s❛✉ ✤➙② t→❝ ❣✐↔ s➩ tr➻♥❤ ❜➔②
ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ✤➸ ❣✐↔✐ q✉②➳t ♠ët sè ❜➔✐ t♦→♥✳


✶✺

❇➔✐ t♦→♥ ✶✳✶ ❳→❝ ✤à♥❤ ❤➔♠ sè f (x) ✤ì♥ ✤✐➺✉ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ ✭✶✳✶✮✳
▲í✐ ❣✐↔✐✿

❱➻ f (x) t❤ä❛ ♠➣♥ ✭✶✳✶✮ ♥➯♥✿ f (x) = a x, ∀x R ợ a = f (1) R tũ
ỵ ❚❛ ❝❤➾ r❛✱ ♥➳✉ f ✤ì♥ ✤✐➺✉ t❤➻✿ f (x) = a x, ∀x ∈ R✳
❚❛ ❝❤ù♥❣ ♠✐♥❤ tr÷í♥❣ ❤đ♣ f ❦❤ỉ♥❣ ❣✐↔♠✱ ❝á♥ tr÷í♥❣ ❤đ♣ f ❦❤ỉ♥❣ t➠♥❣
t❤➻ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü✳
●✐↔ sû✱ f ❦❤ỉ♥❣ ❣✐↔♠ tr➯♥ R✳ ❑❤✐ ✤â✱ a = f (1) ≥ f (0) = 0✳ ❱ỵ✐ ♠é✐

x ∈ R ❜➜t ❦➻✱ t❛ ①➨t ❤❛✐ ❞➣② sè ❤ú✉ t➾ sn ❣✐↔♠ ✈➔ qn t➠♥❣ ❝ò♥❣ ❝â ❣✐ỵ✐ ❤↕♥
❧➔ x✳
❑❤✐ ✤â✱ ∀n ∈ N✱ t❛ ❝â✿

f (sn ) = a sn ,
f (qn ) = a qn .
▼➦t ❦❤→❝✱ f ❦❤æ♥❣ ❣✐↔♠ tr➯♥ R ♥➯♥✿

a sn ≥ f (sn ) ≥ f (x) ≥ f (qn ) = a qn , ∀n ∈ N.
▲➜② ❣✐ỵ✐ ❤↕♥ ❤❛✐ ✈➳ ❦❤✐ n → +∞✱ t❛ ❝â✿

limn→+∞ asn ≥ f (x) ≥ limn→+∞ aqn ,
⇒ ax ≥ f (x) ≥ ax.
❱➟② f (x) = ax✱ ♥❤÷♥❣ x ∈ R ❜➜t ❦➻ ♥➯♥ f (x) = ax, ∀x ∈ R✳

◆❤➟♥ ①➨t✿

✰ ❚ø ❣✐↔ t❤✐➳t f ✤ì♥ ✤✐➺✉ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ ✭✶✳✶✮✱ t❛ ❝â t❤➸ s✉② r❛ f
❧✐➯♥ tö❝ t↕✐ x = 0✳ ❙✉② r❛✱ f (x) = x f (1), ∀x ∈ R✳ ❈→❝❤ ❧➔♠ tr➯♥ s➩ ❦❤→
♥❣➢♥ ❣å♥ ✈➔ rã r➔♥❣ ✤ë❝ ❧➟♣ ❤ì♥ ❧➔ ♥➳✉ t❛ q✉② ✈➲ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ f ✳
✣➙② ❧➔ ❦➳t q✉↔ t ừ t ợ ữỡ tr ❤➔♠ ✈ø❛
❝ë♥❣ t➼♥❤ ✈ø❛ ✤ì♥ ✤✐➺✉✳
✰ ◆➳✉ t❤❛② ❣✐↔ t❤✐➳t f ✤ì♥ ✤✐➺✉ ❜ð✐✿ f (x) > 0, ∀x ∈ R ✈➔ f t❤ä❛ ♠➣♥
✭✶✳✶✮ t❤➻ s✉② r❛ f ❧➔ ❤➔♠ ❦❤æ♥❣ ❣✐↔♠ tr➯♥ R✱ ❞♦ ✤â✿ f (x) = ax, ∀x ∈ R
✈➔ a ≥ 0✳
◆➳✉ f (x2n ) = [f (x)]2n , n ∈ N∗ ✱ s✉② r❛✿ f (x) ≡ 0 ❤♦➦❝ f (x) = x, ∀x ∈

R✳

✶✻

❈á♥ ♥➳✉✱ f (x) ≤ 0, ∀x ≥ 0 s✉② r❛✿ ❤➔♠ f ❦❤æ♥❣ t➠♥❣ tr➯♥ R✱ ❤❛②

f (x) = ax, ∀x ∈ R✱ ✈ỵ✐ a ≤ 0✳

❇➔✐ t♦→♥ ✶✳✷ ❚➻♠ ❝→❝ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ ✭✶✳✶✮

✈➔ ❜à ❝❤➦♥ tr➯♥ ✤♦↕♥ [c, d] ✈ỵ✐ c < d ❜➜t ❦➻✳
▲í✐ ❣✐↔✐✿

●✐↔ sû f ❧➔ ❤➔♠ t❤ä❛ ♠➣♥ ❜➔✐ t♦→♥✳

❉♦ f t❤ä❛ ♠➣♥ ✭✶✳✶✮ ♥➯♥✿ f (x) = ax, ∀x ∈ Q✱ tr♦♥❣ ✤â✿ a = f (1)✳
❚❛ ❝❤ù♥❣ ♠✐♥❤✿ f (x) = ax, ∀x ∈ R✳
❚❤➟t ✈➟②✱ ❧➜② x ∈ R ❜➜t ❦ý✳
❑❤✐ ✤â✱ ợ ộ n N tỗ t rn Q ✭♣❤ö t❤✉ë❝ ✈➔♦ n ✈➔ x✮✱ s❛♦ ❝❤♦✿

nx − d ≤ rn ≤ nx − c.
❙✉② r❛✱ f (nx − rn ) ❜à ❝❤➦♥✱ ❞♦ c ≤ nx − rn ≤ d✳
❚❛ ❝â✿

|f (nx − rn )| = |f (nx) + f (−rn )| = |nf (x) − arn |
= |n(f (x) − ax) + a(nx − rn )| ≥ n|f (x) − ax| − |a(nx − rn )|.
❙✉② r❛✱ |f (nx − rn )| + |a(nx − rn )| ≥ n|f (x) − ax|✳
▼➦t ❦❤→❝✱ |a(nx − rn )| ≥ max{|ac|, |ad|}✱ ✈➔ f (nx − rn ) ❜à ❝❤➦♥ ✈ỵ✐
♠å✐ n ∈ N✳
❉♦ ✤â✱ n|f (x) − ax| ❝ơ♥❣ ❜à ❝❤➦♥ ✈ỵ✐ ♠å✐ n ∈ N✳
✣✐➲✉ ♥➔② ❝❤➾ ①↔② r❛ ❦❤✐✿ f (x) − ax = 0✳
❱➟② f (x) = ax, ∀x ∈ R✳

❇➔✐ t♦→♥ ✶✳✸ ❳→❝ ✤à♥❤ ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥
f

x+y+z
3

=

f (x) + f (y) + f (z)
, ∀x, y, z ∈ R.
3

▲í✐ ❣✐↔✐✿

✣➦t g(x) = f (x) − f (0) t❤➻ g(0) = 0✳
❉♦ ✈➟②✱ ∀x, y, z ∈ R.


✶✼

g

x+y+z
3

=f

x+y+z
3

− f (0)

f (x) + f (y) + f (z)
− f (0)
3
(f (x) − f (0)) + (f (y)f (0)) + (f (z) − f (0))
=
3
=

=

g(x) + g(y) + g(z)
3

g(x)
x
=
✳
3
3
g(y)
y
❈❤♦ x = z = 0✱ s✉② r❛ g
=
✳
3
3
❉♦ ✈➟②✱ ✈ỵ✐ x, y ∈ R t❛ ❝â

❈❤♦ y = z = 0✱ s✉② r❛ g

g(x + y) = g

3x + 3y
3

=

g(3x) + g(3y)
= g(x) + g(y).
3

❱➻ f (x) ❧✐➯♥ tö❝ ♥➯♥ g(x) ❧✐➯♥ tư❝✳
❚❤❡♦ ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② t❛ ❝â✿

g(x) = ax,
❤❛②

f (x) = ax + b, b = f (0)✳

❇➔✐ t♦→♥ ✶✳✹ ❈❤♦ ❤➔♠ sè f ✤ì♥ ✤✐➺✉ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥
f (x + y) = f (x) + f (y) + 2xy, ∀x, y ∈ R

✭✶✳✶✶✮

▲í✐ ❣✐↔✐✿

✣➦t g(x) = f (x) − x2 t❤➻ ♣❤÷ì♥❣ tr➻♥❤✭✶✳✶✶✮ trð t❤➔♥❤

g(x + y) + (x + y)2 = g(x) + g(y) + x2 + y 2 + 2xy.
❙✉② r❛✿ g(x + y) = g(x) + g(y)✳
❚❤❡♦ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝â✿ g(x) = ax ✈➔ f (x) = x2 + ax✳ ❚❤û ❧↕✐✱
t❛ t❤➜② f (x) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✶✮✳


✶✽

❇➔✐ t♦→♥ ✶✳✺ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f : R+ → R
f
▲í✐ ❣✐↔✐✿

1
f (xy)

{0} t❤ä❛ ♠➣♥

= f (x)f (y), ∀x, y > 0

✭✶✳✶✷✮

❚❤❛② y = 1✱ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮ t❛ ✤÷đ❝✿

f

1
f (x)

✣➦t a = f (1) = 0✱ s✉② r❛✿ f
❉♦ ✤â t❛ ❝â✿

f

1
f (xy)

= f (x)f (1).

1
f (x)

= af (x)✳

= af (xy), ∀x, y > 0

✭✶✳✶✸✮

❚ø ✭✶✳✶✷✮ ✈➔ ✭✶✳✶✸✮✱ t❛ ✤÷đ❝✿

af (xy) = f (x)f (y).
❙✉② r❛

f (xy) f (x) f (y)
=
.
.
a
a
a
❱➟② g(xy) = g(x)g(y), ∀x, y > 0,

f (x)
tr♦♥❣ ✤â g(x) =
✳
a
❱➻ f ❧✐➯♥ tö❝ tr➯♥ R+ ♥➯♥ g ❝ơ♥❣ ❧✐➯♥ tư❝ tr➯♥ R+ ✳
❚ø ♠ët ❤➺ q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t❛ ❝â✿
g(x) = xα , ∀x ∈ R+ , 0 = α ∈ R.
❱➟② f (x) = axα , ∀x ∈ R+ ✳
❚❤û ❧↕✐ t❛ t❤➜② f (x) = axα , ∀x ∈ R+ ✱ t❤ä❛ ♠➣♥ ✭✶✳✶✷✮✳

❇➔✐ t♦→♥ ✶✳✻ ❚➻♠ ❝➦♣ f, g ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ (1, +∞) s❛♦ ❝❤♦
f (xy) = xg(y) + yg(x), ∀x, y > 1.
▲í✐ ❣✐↔✐✿

❈❤♦ x = y ✱ t❤❛② ✈➔♦ ✭✶✳✶✹✮ t❛ ❝â✿

f (x2 ) = 2xg(x).

✭✶✳✶✹✮


✶✾

f (x2 )
✳
2x
❚❤❛② ✈➔♦ ✭✶✳✶✹✮ t❛ ❝â

❙✉② r❛✿ g(x) =

xf (y 2 ) yf (x2 )
f (xy) =
+
.
2y
2x
f (xy) 1
=
⇒
xy
2

f (x2 ) f (y 2 )
+ 2
✳
x2
y

❱➟② t❛ ❝â✿

1
√
g ( xy) = (g(x) + g(y)) , ∀x, y > 1
2
u
v
✣➦t x = e , y = e ✳ ❑❤✐ ✤â✱ tữỡ ữỡ ợ


u+v

1
g e e = (g(eu ) + g(ev )) , ∀u, v > 0.
2

✭✶✳✶✺✮

✣➦t h(x) = g(ex )✱ t❛ ❝â✿

h

x+y
2

=

1
(h(x) + h(y)) , ∀x, y > 0.
2

❱➻ g ❧✐➯♥ tö❝ ♥➯♥ h ❧➔ ❤➔♠ ❧✐➯♥ tö❝✳ ⑩♣ ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❏❡♥s❡♥ t❛
✤÷đ❝✿

h(x) = ax + b
⇒ g(x) = alnx + b
tr♦♥❣ ✤â a, b ❧➔ ❝→❝ ❤➡♥❣ sè✳
❱➟②

√
a
ax

.lnx + bx, ∀x > 1.
f (x) = xg( x) = x .lnx + b =
2
2
❚❤û ❧↕✐✱ t❛ t❤➜② f (x), g(x) t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮✳

❇➔✐ t♦→♥ ✶✳✼ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f (x) ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥
f (x + y) = f (x) + f (y) + f (x)f (y)

✭✶✳✶✻✮



ớ

Pữỡ tr tữỡ ữỡ ợ

f (x + y) + 1 = (f (x) + 1).(f (y) + 1).
✣➦t g(x) = f (x) + 1✳ ❚❛ ❝â

g(x + y) = g(x)g(y).
⑩♣ ❞ư♥❣ ❤➺ q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t❛ ✤÷đ❝ g(x) = ax ✳
❱➟②✿ f (x) = ax − 1✳

❇➔✐ t♦→♥ ✶✳✽ ❈❤♦ a ∈ R✱ t➻♠ t➜t ❝↔ ❤➔♠ ❧✐➯♥ tö❝ f : R → R t❤ä❛ ♠➣♥
f (x − y) = f (x) − f (y) + axy, ∀x, y ∈ R

✭✶✳✶✼✮

▲í✐ ❣✐↔✐✿

❚❤❛② x = 1, y = 0 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✼✮✱ t❛ ❝â✿

f (1) = f (1) − f (0).
❙✉② r❛ f (0) = 0✳
❚❤❛② x = 1, y = 1 ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ t ữủ a = 0
ợ a = 0 t số f ổ tỗ t
t q ❤➺ ❤➔♠

f (x − y) = f (x) − f (y), ∀x, y ∈ R.
❚❛ ❝â✿ f (x) = f (x + y − y) = f (x + y) − f (y), ∀x, y ∈ R✳
❙✉② r❛ f (x + y) = f (x) + f (y), ∀x, y ∈ R✳
⑩♣ ❞ư♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t❛ ❝â✿ f (x) = cx, ∀x ∈ R, c ❧➔
❤➡♥❣ sè✳

❇➔✐ t♦→♥ ✶✳✾ ❚➻♠ t➜t ❝↔ ❤➔♠ sè f : R → R ❧✐➯♥ tö❝ ✈➔ t❤ä❛ ♠➣♥
f (x + y) + f (z) = f (x) + f (y + z), ∀x, y, z ∈ R

✭✶✳✶✽✮



ớ

ờ ữỡ tr tữỡ ữỡ ợ

f (x + y) − f (x) = f (y + z) − f (z), ∀x, y, z ∈ R

✭✶✳✶✾✮

❉♦ ✈➳ ♣❤↔✐ ❝õ❛ ✭✶✳✶✾✮ ❦❤æ♥❣ ❝❤ù❛ x ♥➯♥ ✈➳ tr→✐ ❝õ❛ ✭✶✳✶✾✮ ❦❤ỉ♥❣ ♣❤ư
t❤✉ë❝ ✈➔♦ x✳
✣➦t g(y) = f (x + y) − f (x)✱ t❛ ❝â✿

f (x + y) − f (x) = g(y), ∀x, y ∈ R.
❱ỵ✐ x = 0✱ t❛ ❝â✿

f (y) = g(y) + f (0) = g(y) + a, a = f (0).
❑❤✐ ✤â ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✽✮ trð t❤➔♥❤✿

g(x + y) + a + g(z) + a
= g(x) + a + g(y + z) + a.
⇔ g(x + y) = g(x) + (g(z + y) − g(z)).

✭✶✳✷✵✮

❚❛ ❝â✿

g(z + y) − g(z) = f (x + y) − a − (f (z) − a)
= f (z + y) − f (z)
= g(y), ∀y, z ∈ R
❉♦ ✤â ❤➺ t❤ù❝ ✭✶✳✷✵✮ trð t❤➔♥❤✿

g(x + y) = g(x) + g(y), ∀x, y ∈ R

✭✶✳✷✶✮

P❤÷ì♥❣ tr➻♥❤ ✭✶✳✷✶✮ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥➯♥ t❛
❝â✿

g(x) = cx, ∀x ∈ R,
c ❧➔ ❤➡♥❣ sè✳
❱➟② ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦ ❧➔✿ f (x) = cx + a.