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♥❣ ♣❤→♣ ❚♦→♥ ❝➜♣ ▼➣ sè✿ ✻✵ ✹✻ ✵✶ ✶✸ ▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈ ◆●×❮■ ❍×❰◆● ❉❼◆ ❑❍❖❆ ❍➴❈✿ ❚❙✳ ◆●❯❨➍◆ ✣➐◆❍ ❇➐◆❍ ❚❍⑩■ ◆●❯❨➊◆✱ ✷✵✶✼ ✐ ▲❮■ ❈❷▼ ❒◆ ▲✉➟♥ ✈➠♥ ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ tr÷í♥❣ ✣↕✐ ữợ sỹ ữợ tr trå♥❣ ❜➔② tä ❧á♥❣ ❦➼♥❤ trå♥❣ ✈➔ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐ ❚❙✳◆●❯❨➍◆ ✣➐◆❍ ❇➐◆❍✱ t❤➛② ✤➣ t➟♥ t➻♥❤ ❝❤➾ ữợ t ❦✐➺♥ t❤✉➟♥ ❧ñ✐ ❝❤♦ t→❝ ❣✐↔ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥✳ ◗✉❛ ❜↔♥ ❧✉➟♥ ✈➠♥ ♥➔②✱ t→❝ ❣✐↔ ①✐♥ ❣û✐ ❧í✐ ❝↔♠ ì♥ tỵ✐ ❝→❝ t❤➛② ❝ỉ tr♦♥❣ tr÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤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 ữớ q t➙♠✱ ✤ë♥❣ ✈✐➯♥ ✈➔ ❣✐ó♣ ✤ï ✤➸ t→❝ ❣✐↔ ❝â t❤➸ ❤♦➔♥ t❤➔♥❤ ❧✉➟♥ ✈➠♥ ❝õ❛ ♠➻♥❤✳ ❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥✦ ❚❤→✐ ◆❣✉②➯♥✱ t❤→♥❣ ✵✻ ♥➠♠ ✷✵✶✼ ❍å❝ ✈✐➯♥ ❇ị✐ ❚❤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♥❣ ❞ö♥❣✳ ✷✺ ✷✳✶ ❚✐➳♣ ❝➟♥ ❣✐→ 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❤❛♠ ❦❤↔♦✳ ✺✺ ✺✼ ✶ ▼Ð ✣❺❯ ỵ t ởt ữỡ tr ữủ ♥❤✐➲✉ ♥❣÷í✐ ❜✐➳t ✤➳♥ ✈➔ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝ì ❜↔♥ tr ỵ tt ữỡ tr ữỡ tr ❈❛✉❝❤②✳ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❧➽♥❤ ✈ü❝ ❤❛② ✈➔ ❦❤â ❝õ❛ t♦→♥ ❤å❝ ❝➜♣✱ ♥â õ ự tr ỵ tt ữỡ tr ✈➔ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ t♦→♥ ❤å❝ ✈➔ ❦❤♦❛ ❤å❝ ỗ t ự t➼❝❤✱ ❣✐↔✐ t➼❝❤ ♣❤ù❝✱ ①→❝ ①✉➜t t❤è♥❣ ❦➯✱ ❣✐↔✐ t➼❝❤ ❤➔♠✱ ✤ë♥❣ ❧ü❝ ❤å❝✱ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ❝ì ❤å❝ ❝ê ✤✐➸♥✱ ❝ì ❤å❝ t❤è♥❣ ❦➯ ✈➔ ❦✐♥❤ t➳ ❤å❝✳ 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❛ æ♥❣ tø ♥➠♠ ✶✽✵✾✱ ♥❤÷♥❣ sü ♥❣❤✐➯♥ ❝ù✉ ♥➔② ❦❤æ♥❣ ❝❤➦t ❝❤➩ ✈➔ ❝ơ♥❣ ❦❤ỉ♥❣ rã r➔♥❣✳ ❚rð ❧↕✐ ♥❤ú♥❣ trữợ ỳ t õ t t t tr♦♥❣ s→❝❤ ❝õ❛ ▲❡❣❡♥❞r❡✱ ð ♣❤➛♥ ❞➔♥❤ ❝❤♦ sü ♥❣❤✐➯♥ ❝ù✉ t➾ sè ❞✐➺♥ t➼❝❤ ❝õ❛ ❝→❝ ❤➻♥❤ ❝❤ú ♥❤➟t✱ ❝↔ ù♥❣ ❞ư♥❣ ✈➔ ♣❤➙♥ t➼❝❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t✉② ♥❤✐➯♥ ❝❤ó♥❣ ✈➝♥ ❝❤÷❛ ❝❤➦t ❝❤➩ ✈➔ ❦❤ỉ♥❣ rã r➔♥❣✳ ❉♦ ✤â ♥â ✤➣ t❤✉ ❤ót sü ❝❤ó þ ❝õ❛ ❝→❝ t→❝ ❣✐↔ tr♦♥❣ ❦❤♦↔♥❣ t❤í✐ ❣✐❛♥ ❞➔✐✳ ❑❛♥♥❛♣♣❛♥ ✤➣ ✈✐➳t✿ ✏❈→❝ ♥❤➔ ♥❣❤✐➯♥ ❝ù✉ ✤➣ ✤❛♠ ♠➯ ♥❤ú♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❬P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ✸ ❦✐➸✉ t÷ì♥❣ ✤÷ì♥❣❪✱ ✈➔ sü ↔♦ t÷ð♥❣ ♥➔② s➩ t✐➳♣ tư❝ ✈➔ ❞➝♥ ✤➳♥ ♥❤✐➲✉ t❤➔♥❤ q✉↔ t❤ó ✈à ❤ì♥✳✑ ữợ ự ữỡ 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✱ ✈ỵ✐ ♠å✐ x ∈ R✱ ✈➔ t❤ü❝ t➳ ♥➔② ✤➣ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜➡♥❣ ♥❤✐➲✉ ❝→❝❤✳ ❱➼ ❞ư✱ ❈❛✉❝❤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 ❦❤✐ ❦❤ỉ♥❣ ❝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 s t ữợ ởt ữợ ❧➜② ♠✐➲♥ ①→❝ ✤à♥❤ ✈➔ ♠✐➲♥ ❣✐→ trà ❝õ❛ f t❤➔♥❤ ❝→❝ ♥❤â♠ ❝õ❛ ❧♦↕✐ ♥➔♦ ✤â✱ ✈➼ ❞ö ✭❝♦♠♣❛❝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❤➻ ♥â ♣❤↔✐ ❧✐➯♥ tử ú ỵ t õ ữỡ tr t❛ ♥â✐ ❧➔ ❤➔♠ t❤✉➛♥ ♥❤➜t ❤♦➦❝ ❝ë♥❣ t➼♥❤✮✳ ▼ët ữợ ủ 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 ❝õ❛ ♠✐➲♥ ①→❝ ✤à♥❤ ❝❤➼♥❤ t❤ù❝✱ ✈➼ ❞ö ♠ët t ỗ ũ t ữủ ❙ü ❜✐➳♥ ✤ê✐ ♥➔② ❧➔ ✤➸ t❤❛② ✤ê✐ ♠✐➲♥ ❣✐→ trà ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✱ ✈➼ ❞ư✱ ❣✐↔ sû r➡♥❣ f tọ ữỡ tr ợ (x, y) t❤✉ë❝ ✈➔♦ t➟♣ ❝♦♥ ❝õ❛ R2n ✱ ✈➼ ❞ö ✤❛ t↕♣ ✭✈➔ f ❝â t❤➸ ✤÷đ❝ ✤à♥❤ ♥❣❤➽❛ tr➯♥ t♦➔♥ ❦❤æ♥❣ ❣✐❛♥ ❤♦➦❝ tr➯♥ t➟♣ ❝♦♥ ❝õ❛ ♥â✮✳ ❚r♦♥❣ t➜t ❝↔ ❝→❝ tr÷í♥❣ ❤đ♣ ♥➔②✱ t❛ ❝â t❤➸ ❦➳t sỹ tỗ t ổ t t ữỡ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✭♠➦❝ ❞ò ❝→❝ ✤✐➲✉ ❦✐➺♥ ✤➲✉ ♠↕♥❤✮ ❤♦➦❝ ✭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 ❧➔ ❝ỉ♥❣ ✸ ❝ư ✤➸ ❣✐↔✐ q✉②➳t r➜t ♥❤✐➲✉ ❜➔✐ t♦→♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❤❛② ✈➔ ❦❤â✱ ♥â ①✉➜t ❤✐➺♥ ♥❤✐➲✉ tr♦♥❣ ❝→❝ ✤➲ t❤✐ ❤å❝ s✐♥❤ ọ tr ữợ qố t tữớ ởt t❤→❝❤ t❤ù❝ ✤è✐ ✈ỵ✐ ❤å❝ 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❝ ✈➔ sỹ tữỡ ố õ ố ợ t➟♣ ❝♦♥ ❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❊✉❝❧✐❞❡ ✤❛ ❝❤✐➲✉✳ ▼ët sè ợ ữủ tr ữ ♠ët ♣❤÷ì♥❣ tr➻♥❤ tr♦♥❣ ✤â ♠ët sè ♠ơ ♣❤ù❝ t↕♣ ❝→❝ ❤➔♠ ❝❤÷❛ ❜✐➳t✳✳✳✳ ❈→❝ ♣❤➙♥ t➼❝❤ ✤÷đ❝ ♠ð rë♥❣ ✤➳♥ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ t ự ỵ tt tr ỗ ữù tự t ❝❤♦ ❤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➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ự r q ợ t ữợ ✈➲ ❝→❝ ù♥❣ ❞ư♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♥â✳ ✺✳ ❇è ❝ö❝ ❧✉➟♥ ✈➠♥✳ ✹ ❚→❝ ❣✐↔ t✐➳♥ ❤➔♥❤ ♥❣❤✐➯♥ ❝ù✉ ❤❛✐ ♥ë✐ 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♥❣ ❞ö♥❣✳ ✷✳✶✳ ❚✐➳♣ ❝➟♥ ❣✐→ trà ❜❛♥ ✤➛✉✳ ✷✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② tr➯♥ ♠✐➲♥ ❤↕♥ ❝❤➳✳ ✷✳✸✳ ▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤②✳ ✷✳✹✳ ▼ët sè ✈➼ ❞ư ♠✐♥❤ ❤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❛ õ t ỹ ữủ ởt số ữợ ữ s❛✉✿ ✶✳ ❚❤➳ ❝→❝ ❣✐→ trà ❜✐➳♥ ♣❤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➼♥❤ ❧✐➯♥ tö❝ ❝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 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, ữủ t ỵ ✶✳✶ ❍➔♠ sè ❧✐➯♥ tö❝ f (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ f (x) = ax, ∀x ∈ R, tr♦♥❣ ✤â✱ a ❧➔ ❤➡♥❣ sè tũ ỵ F ✤÷❛ ✤➳♥ ✤➥♥❣ t❤ù❝✿ F (x) = f (s) − f (t) = f (x) ❈✉è✐ ❝ò♥❣✱ ✤➸ t❤➜② f ❝ë♥❣ t➼♥❤✱ ❝❤♦ (x1 , x2 ) ∈ G2 ❜➜t S s G tỗ t s1 , s2 , t1 , t2 ∈ S ✱ t❤ä❛ ♠➣♥ ❝→❝ q✉❛♥ ❤➺✿ x1 = s1 − t1 , x2 = s2 − t2 , s1 + s2 ∈ S, t1 + t2 ∈ S ✣✐➲✉ ♥➔② ✈➔ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ G ✤÷❛ ✤➳♥✿ x1 + x2 = (s1 + s2 ) − (t1 + t2 ) ∈ S − S ✣✐➲✉ ♥➔②✱ ✤à♥❤ ♥❣❤➽❛ ❝õ❛ F ✱ t➼♥❤ ❣✐❛♦ ❤♦→♥ ❝õ❛ H ✈➔ ✭✷✳✻✮ s✉② r❛ ✤✐➲✉ ❦❤➥♥❣ ✤à♥❤ ✤÷đ❝ ②➯✉ ❝➛✉✿ F (x1 + x2 ) = f (s1 + s2 ) − f (t1 + t2 ) = f (s1 ) − f (t1 ) + f (s2 ) − f (t2 ) = F (x1 ) + F (x2 ) ỵ sỷ S S (S + S) ⊆ A✳ ✣➦t ♠ët s✐➯✉ ❧➟♣ ♣❤÷ì♥❣ ❝❤♦✿ ⊆ Rn s✐♥❤ ♠↕♥❤ Rn ✳ ❈❤♦ A ⊆ Rn t❤ä❛ ♠➣♥✿ f : A → R ✈➔ ❣✐↔ sû f t❤ä❛ ♠➣♥ ✭✷✳✻✮✳ ◆➳✉ S ❝❤ù❛ I tr õ eif ữủ t tỗ t↕✐ c ∈ Rn ✱ s❛♦ f (x) = c.x, ∀x ∈ S ✳ ❈❤ù♥❣ ♠✐♥❤✿ ❇➡♥❣ ❇ê ✤➲ ✷✳✹✱ tỗ t ởt t F : Rn R t❤ä❛ ♠➣♥✿ F (x) = f (x) ✈ỵ✐ ♠å✐ x ∈ S ✳ ❱➻ F ❝ë♥❣ t➼♥❤ ✈➔ eiF = eif ✤♦ ✤÷đ❝ tr➯♥ I ✱ t❛ s✉② r❛ tứ ỵ r tỗ t c Rn s❛♦ ❝❤♦ F (x) = c.x ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❑❤➥♥❣ ✤à♥❤ ✤÷đ❝ s✉② r❛ ✈➻ f (x) = F (x) ✈ỵ✐ ♠å✐ x ∈ S ✳ ◆❤➟♥ ①➨t ✷✳✺ ❈→❝ ❤➻♥❤ ♠➝✉ ✈➲ ❝→❝ t➟♣ ❝♦♥ S Rn s Rn ỗ ởt s✐➯✉ ❧➟♣ ♣❤÷ì♥❣ ❧➔ ❝→❝ ❣✐❛♦ ❧➝♥ ♥❤❛✉ ❝õ❛ ❝→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥ trü❝ ❣✐❛♦✱ ❝→❝ ♥û❛ ❦❤æ♥❣ ❣✐❛♥✱ ✈➔ t ỗ ởt t s Rn ❝â ♣❤➛♥ tr♦♥❣ ❦❤→❝ ré♥❣✳ ✹✹ ❚❛ ❝â✱ ♥û❛ ♥❤â♠ S ⊆ Rn ✭tù❝ ❧➔ S + S ⊆ S ✮ ♠➔ s✐♥❤ r❛ Rn ✳ ✣➦❝ ❜✐➺t✱ S : ∪∝ m=1 [2m, 3m] ✤õ ♠↕♥❤ s✐♥❤ r❛ R✳ ❈ô♥❣ ❝â ♥❤✐➲✉ ✈➼ ❞ư ✈➲ t➟♣ ❝♦♥ S ✈ỵ✐ ❣✐❛♦ ❦❤→❝ ré♥❣ ♠➔ s✐♥❤ r❛ Rn ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ♥û❛ ♥❤â♠✳ ▼ët ✈➼ ❞ư ✤ì♥ ❣✐↔♥ ❧➔ sü ❞à❝❤ ❝❤✉②➸♥ ❝õ❛ ❣â❝ ♣❤➛♥ t÷ ❜ð✐ ♠ët ✈➨❝tì ❦❤ỉ♥❣ t❤✉ë❝ ✈➔♦ ❣â❝ ♣❤➛♥ t÷✳ ◆❤÷♥❣ ❦❤ỉ♥❣ ❝â ❝→❝ ✈➼ ❞ư ♥❣♦↕✐ ❧❛✐ ♥❤÷✿ m m n n S : ∪∝ m=1 Sm ✱ ð ✤â Sm := [10 , 5.10 ) , ∀m ∈ N✱ ✭❱➻✿ ❱ỵ✐ x, y ∈ R ✱ ❧➜② ≤ m ∈ N s❛♦ ❝❤♦ x + y| < 10m−1 ❀ ❝❤å♥ pm := (2.10m )nk=1 ❧➔ ♥❤ú♥❣ ✈➨❝tì ❝õ❛ 2.10m ✱ t❤➻✿ x = (pm + x) − pm ∈ Sm − Sm ✱ y = (pm + y) − pm ∈ Sm − Sm , ✈➔ x + y = (2pm + x + y) 2pm Sm Sm ỵ tữỡ tỹ ự ỵ õ t ✤÷đ❝ sû ❞ư♥❣ ✤➸ ❞➵ ❞➔♥❣ ♠ð rë♥❣ ♥â ❤ì♥ ỳ ữ ỵ t t ỵ S ⊆ Rn✳ ❈❤♦ A ⊆ Rn t❤ä❛ ♠➣♥ S ∪(S +S) ⊆ A✳●✐↔ sû f :A→R t❤ä❛ ♠➣♥ ✭✷✳✻✮✳●✐↔ sû ❧➔ ✤♦ ✤÷đ❝ tr➯♥ I S ❝❤ù❛ ♠ët ❤➻♥❤ ữỡ I eif tỗ t ởt ❝ë♥❣ t➼♥❤ ✭❦❤æ♥❣ ❝➛♥ ❧➔ ❞✉② ♥❤➜t✮ F : Rn → R✱ s❛♦ ❝❤♦✿ F (x) = f (x), ∀x ∈ S f (x) = cx, ∀x ∈ S ✳ t tỗ t c Rn s ỵ ữủ t tr tr t õ t ✤÷đ❝ sû ❞ư♥❣✳ ❱➻ ✈➟②✱ t❛ ❝â t❤➸ ❝❤♦ S tỵ✐ ❜➜t ❦ý ❦❤♦↔♥❣ ♥➔♦ tr➯♥ R ❝â ✵ ❧➔ ✤✐➸♠ ❞➼♥❤ ❝õ❛ ♥â ✈➔ A := S + S ✳ ❚❛ ❝â t❤➸ ❝❤♦ S tỵ✐ ♠ët q✉↔ ❝➛✉ ❝➜♣ sè ♥❤➙♥ ❝â ❝❤✐➲✉ ✤✐➸♠ ❣è❝ ❧➔ t➙♠ ❝õ❛ ♥â ✭✈➔ A := S + S ✮ ♥❤÷♥❣ ✤✐➲✉ tỹ sỹ r tứ ỵ t t ✭tr÷í♥❣ ❤đ♣ ♠ð rë♥❣ ❝õ❛ ♠ët ❤➔♠ ❧✐➯♥ tư❝✱ ❝ë♥❣ t ữủ tr ởt ỵ S Rn ởt t ỗ ❝â ❣✐❛♦ ❦❤→❝ ré♥❣✳ ❈❤♦ A ⊆ Rn t❤ä❛ ♠➣♥✿ S ∪ (S + S) ⊆ A✳ ●✐↔ sû f : A → R t❤ä❛ ♠➣♥ ✭✷✳✻✮✳ if n ◆➳✉ e ữủ tr S t tỗ t c ∈ R ✱ s❛♦ ❝❤♦ f (x) = cx, ∀x ∈ S ✳ ❈❤ù♥❣ ♠✐♥❤ ✹✺ ❈❤♦ u ∈ S + S tũ ỵ t u = s + t ✈ỵ✐ ♠ët ✈➔✐ s, t ∈ S ✈➔ õ t ỗ t x = y = u s+t = ∈ S ✈➻ S ❧➔ 2 u tr♦♥❣ ✭✷✳✻✮✱ ✤✐➲✉ ♥➔② s✉② r❛ r➡♥❣✿ f (u) u =f , ∀u ∈ S + S 2 ✣➦❝ ❜✐➺t ✤✐➲✉ ♥➔② ✤ó♥❣ ❝❤♦ u := x + y, x, y ∈ S ✤➣ ✤÷đ❝ ❝❤♦✳ ❉♦ ✤â✱ f x+y = f (x + y) f (x) + f (y) = , 2 ð ♣❤÷ì♥❣ tr➻♥❤ ❝✉è✐ ✤÷đ❝ s✉② r❛ tø ✭✷✳✻✮✳ ❙✉② r❛ tỗ t c Rn b R s ❝❤♦✿ f (x) = cx + b, ∀x ∈ S ◆❤í sü ✤â♥❣ ❦➼♥ ❝õ❛ ❜✐➸✉ ❞✐➵♥ ✭✷✳✻✮✱ t❛ s✉② r❛ b = 0✳ ✷✳✸ ▼ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❚r♦♥❣ ♠ư❝ ♥➔②✱ t→❝ ❣✐↔ tr➻♥❤ ❜➔② ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ ♥â ❝â ♥❤ú♥❣ ù♥❣ ❞ö♥❣ tr♦♥❣ ❣✐↔✐ q✉②➳t ♠ët sè ❜➔✐ t♦→♥✳ ✷✳✸✳✶ P❤÷ì♥❣ tr➻♥❤ ❏❡♥s❡♥ ▲➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝â ❞↕♥❣✿ f x+y = f (x) + f (y) ✭✷✳✼✮ ❚r♦♥❣ ✤â✿ x, y ∈ Rn ❤♦➦❝ ởt t Rn ỵ S ởt t ỗ Rn sû ♣❤➛♥ tr♦♥❣ ❝õ❛ f : S → R t❤ä❛ ♠➣♥ ✭✷✳✼✮ ✈ỵ✐ ♠å✐ x, y ∈ S ✳ eif S t tỗ t c Rn b ∈ R✱ s❛♦ ❝❤♦✿ f (x) = cx + b ợ õ rộ sỷ ữủ tr➯♥ x ∈ S✳ ✹✻ ❈❤ù♥❣ ♠✐♥❤✿ x+y ∈ S ✱ ♥➯♥ ✭✷✳✶✮ ❤♦➔♥ t♦➔♥ ✤÷đ❝ ①→❝ ✤à♥❤✳ ❑❤✐ õ tỗ t ởt g : Rn R tọ ởt số S ỗ b ∈ R s❛♦ ❝❤♦✿ f (x) = g(x) + b ✈ỵ✐ ♠å✐ x ∈ S ✳ ❱➻ eif ❧➔ ✤♦ ✤÷đ❝ tr➯♥ S ✈➔ ♣❤➛♥ tr♦♥❣ ❝õ❛ S ❦❤→❝ ré♥❣✱ eif ✤♦ ✤÷đ❝ tr➯♥ ♠ët s✐➯✉ ❧➟♣ ♣❤÷ì♥❣ ✤÷đ❝ ❝❤ù❛ tr♦♥❣ S ✳ ❱➻ ✈➟②✱ eig = e−bi ef ✤♦ ✤÷đ❝ tr➯♥ ❤➻♥❤ s✐➯✉ ❧➟♣ ♣❤÷ì♥❣ ♥➔②✳ ❱➻ t❤➳ tỗ t c Rn s g(x) = cx✱ ✈ỵ✐ x ∈ Rn ✳ ❱➟②✱ f (x) = cx + b ✈ỵ✐ ♠å✐ x ∈ S ✱ ✈➔ ♠ët sü ❦✐➸♠ tr❛ ♥❣❛② ❧➟♣ tù❝ ❝❤➾ r❛ ❤➔♠ ♥➔② t❤ä❛ ♠➣♥ ✭✷✳✼✮✱ ✈ỵ✐ ♠å✐ (x, y) ∈ S ✳ ✷✳✸✳✷ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ♥❤➙♥ t➼♥❤ ✣â ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣✿ ✭✷✳✽✮ f (x + y) = f (x)f (y) tr♦♥❣ ✤â✱ ❝➦♣ (x, y) t❤✉ë❝ ♠ët t➟♣ ❝♦♥ ❝õ❛ R2n ✳ ▼ët ♥❣❤✐➺♠ ❤✐➸♥ ♥❤✐➯♥ s✉② r❛ tø ✭✷✳✽✮ ❧➔ f ≡ 0✳ ◆❤÷ ✤➣ ❜✐➳t✱ ❦❤✐ ①➨t tr♦♥❣ t♦➔♥ ❜ë ❦❤æ♥❣ ❣✐❛♥✱ ♥➳✉ f (x0 ) = ð ♠ët sè ✤✐➸♠ x0 ∈ Rn ✱ t❤➻✿ f (x) = f (x − x0 ).f (x0 ) = 0, ∀x ∈ Rn x , ∀x ∈ Rn ✱ ✤✐➲✉ ✤â ①↔② r❛ ❦❤✐ f ❦❤æ♥❣ ❧➔ ❤➔♠ ❤➡♥❣ ✵✱ t❤➻ f (x) > 0, ∀x✳ ❚✉② ♥❤✐➯♥✱ ❦❤✐ ❝➦♣ (x, y) ❝❤➾ ♣❤ö t❤✉ë❝ ✈➔♦ ♠ët t➟♣ ❝♦♥ ❝õ❛ R2n t❤➻ ❱➻ ❤➔♠ ❝ë♥❣ t➼♥❤✿ f (x) = f ✤✐➲✉ ✤â ❦❤æ♥❣ ❝❤➾ r❛ rã r➔♥❣ t↕✐ s❛♦ ♠ët ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✽✮ õ t ữỡ ỵ t t õ ✈➲ t➼♥❤ ❞÷ì♥❣ ❝õ❛ f ✤÷đ❝ ❣✐↔ ✤à♥❤ tr♦♥❣ ✤➲ t ỵ S Rn A ⊆ Rn✒ t❤ä❛ ♠➣♥✿ S ∪(S + S) ⊆ A✳ ●✐↔ sû✱ f :A→R ❱➔ ❣✐↔ sû✱ S ❧➔ ♠ët ữỡ t ợ ởt t ỗ õ tr rộ (x, y) ∈ S ✳ fi ❧➔ ✤♦ ✤÷đ❝ ✹✼ tr➯♥ ♠➔ S✱ fi ❤♦➦❝ S ✤õ ♠↕♥❤ ✤➸ s✐♥❤ r❛ ữủ õ tỗ t Rn õ ❝❤ù❛ ♠ët ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣ c ∈ Rn I s❛♦ ❝❤♦✿ f (x) = ecx , ∀x ∈ S ❈❤ù♥❣ ♠✐♥❤✿ ❱➻ f ❧➔ ❞÷ì♥❣✱ ❤➔♠ sè g : A → R ①→❝ ✤à♥❤ ❜ð✐ g := ln(f ) ❝ô♥❣ ①→❝ ✤à♥❤ tr➯♥ A ✈➔ ❧➜② ❧♦❣❛r✐t ❝õ❛ ✭✷✳✽✮ t❛ t❤➜② r➡♥❣ g ❧➔ ❝ë♥❣ t➼♥❤✳ ❱➻ f i = exp(i ln(f )) = eig ✤÷đ❝ ❣✐↔ sû ❧➔ ✤♦ ữủ tr S I ứ ỵ ỵ s r tỗ t c Rn s❛♦ ❝❤♦✿ g(x) = cx, ∀x ∈ Rn ❱➻ ✈➟②✿ f (x) = ecx , ∀x ∈ S ✳ ❱➟②✱ f t❤ä❛ ♠➣♥ ✭✷✳✽✮✳ ✷✳✸✳✸ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❧✉➙♥ ♣❤✐➯♥ ▲➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝â ❞↕♥❣✿ (f (x + y))2 = (f (x) + f (y))2 ✭✷✳✾✮ tr♦♥❣ ✤â✱ x, y ∈ Rn ❜➜t ❦ý✳ ❍✐➸♥ ♥❤✐➯♥✱ ❜➜t ❦ý ♥❣❤✐➺♠ ♥➔♦ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✭✷✳✶✮ ✤➲✉ ❧➔ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✾✮✱ ♥❤÷♥❣ ♥❣÷đ❝ ❧↕✐ ❧➔ ❦❤ỉ♥❣ ✤ó♥❣❀ ❜ð✐ ✈➻✱ ✤➛✉ t✐➯♥ t❛ ❝❤➾ ❝â t❤➸ s✉② r❛ q✉❛♥ ❤➺ f (x + y) = ±(f (x) + f (y)) ✭ð ✤â✱ ❞➜✉ ♣❤ö t❤✉ë❝ ✈➔♦ ❝➦♣ (x, y)✮✳ ❚✉② ♥❤✐➯♥✱ ♠å✐ ♥❣❤✐➺♠ ❝õ❛ ✭✷✳✾✮ ✤÷đ❝ ①→❝ ✤à♥❤ tr➯♥ ♠ët ♥û❛ ♥❤â♠ S ❝õ❛ Rn ♣❤↔✐ ❧➔ ❝ë♥❣ t ứ ỵ t t s r ỵ s ỵ sû S ❧➔ ♠ët ♥û❛ ♥❤â♠ ❝õ❛ Rn ♠➔ s✐♥❤ r❛ Rn✳ ●✐↔ sû ♠➔ eif f :S→R t❤ä❛ ♠➣♥ tỗ t ữỡ ữủ t tỗ t c Rn s f (x) = cx, ∀x ∈ S I⊂S ✹✽ ✷✳✸✳✹ P❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r✳ ✣â ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tê♥❣ q✉→t ❤â❛ ♠ët ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ❧✐➯♥ q✉❛♥ ✤➳♥ ❜❛ ❤➔♠ ❜✐➳♥ sè✿ f (x + y) = g(x) + h(y), ✭✷✳✶✵✮ tr♦♥❣ ✤â (x, y) ∈ R2n ❤♦➦❝ ♠ët t R2n ỵ t t rở t q ữủ t ố ợ t ữủ ỵ S Rn ❧➔ ♠ët ♥û❛ ♥❤â♠✱ t❤ä❛ ♠➣♥ ∈ S ✳ ●✐↔ sû✿ f : S −→ R✱ g : S −→ R✱ h : S −→ R (x, y) ∈ S ✳ ●✐↔ sû r➡♥❣ S s✐♥❤ r❛ Rn t❤ä❛ ♠➣♥✭✷✳✶✵✮✱ ✈ỵ✐ ♠å✐ ♠➔ sè ♠ơ ♣❤ù❝ ❝õ❛ ♠ët tr ữủ õ tỗ t ✈➔ ❤➡♥❣ sè a, b ∈ R h(x) = c.x + b ✈ỵ✐ ♠å✐ s❛♦ ❝❤♦✿ I c ∈ Rn ✈➔ ♥â ❝❤ù❛ ♠ët ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣ f (x) = c.x + a + b, g(x) = c.x + a ✈➔ x ∈ S ❈❤ù♥❣ ♠✐♥❤✿ ❈❤♦ a := g(0), b := h(0) ✈➔ ❝❤♦ p : S −→ R ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿ p(x) := f (x) − a − b, ✈ỵ✐ ♠å✐ x ∈ S ✳ ❍✐➸♥ ♥❤✐➯♥✱ t❛ ❝â✿ f (x) = p(x)+a+b, g(x) = p(x)+a, h(x) = p(x)+b✱ ✈ỵ✐ ♠å✐ x ∈ S ✈➔ p ❧➔ ❝ë♥❣ t➼♥❤✳ ❱➻ sè ♠ô ♣❤ù❝ ❝õ❛ ✭✷✳✶✮✱ ❝→❝ ❤➔♠ f, g ❤♦➦❝ h ✤÷đ❝ ❣✐↔ sû ❧➔ ✤♦ ✤÷đ❝ ❝õ❛ t➙♠ ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣✱ ❜✐➸✉ ❞✐➵♥ tr➯♥ s✉② r❛ r➡♥❣ eip ❧➔ ✤♦ ✤÷đ❝ ❝õ❛ t➙♠ ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣✳ ❚ø ✤â✱ ♠ët ♥û❛ ♥❤â♠ s✐♥❤ r❛ ♠ët ♥❤â♠ ❣✐❛♦ ❤♦→♥ t❤ü❝ sü ♠↕♥❤ t↕♦ r❛ ♥â ✭✣➙② ❧➔ ❦➳t q✉↔ trü❝ t✐➳♣ tø ✤à♥❤ ♥❣❤➽❛ ✷✳✶ ✈➔ ✈➻ S + S S ỵ ợ A := S s r tỗ t c Rn s❛♦ ❝❤♦✿ p(x) = c.x✱ ✈ỵ✐ ♠é✐ x ∈ S✳ ❑❤➥♥❣ ✤à♥❤ s❛✉ ❞➵ ❞➔♥❣ ❦✐➸♠ tr❛ r➡♥❣ ♥❤➟♥ ✤÷đ❝ ❜ë ❜❛ f, g, h t❤✉ ✤÷đ❝ ❧➔ ♥❣❤✐➺♠ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✵✮✳ ✹✾ ✷✳✸✳✺ ❚➼♥❤ ê♥ ✤à♥❤ ❳➨t ❞↕♥❣ ①➜♣ ①➾ ❜✐➳♥ t❤➸ ❝õ❛ ✭✷✳✶✮ |f (x + y) − f (x) − f (y)| ≤ ✈ỵ✐ ♠å✐ x, y Rn ợ ởt số ữỡ ởt ❤➔♠ f : Rn −→ R t❤ä❛ ♠➣♥ ✭✷✳✶✶✮ ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ①➜♣ ①➾ ❝ë♥❣ t➼♥❤ ❤♦➦❝ ♠ët ❤➔♠ ✲ ❝ë♥❣ t➼♥❤✳ ❇➜t ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✶✮ ❧➔ ♠ët ❜✐➳♥ t❤➸ ✤÷đ❝ ❧➔♠ ♥❤✐➵✉ ❝õ❛ ✭✷✳✶✮ ✈➔ t❛ ❝â t❤➸ ❜✐➳t ❝â ❤❛② ❦❤æ♥❣ ❤➔♠ ✲ ❝ë♥❣ t➼♥❤ ❜➜t ❦ý ❧➔ ✤÷đ❝ ①→♦ trë♥ ➼t ❝õ❛ ♠ët ❤➔♠ ❝ë♥❣ t➼♥❤ t❤✉➛♥ t➼♥❤✳ ❍②❡rs ✤÷❛ r❛ ❝➙✉ tr↔ ❧í✐ ❝õ❛ ✈➜♥ tr õ tỗ t t ởt g : Rn −→ R t❤ä❛ ♠➣♥ ✭✷✳✶✮ ✈➔ |f (x) − g(x)| ≤ ✱ ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❍➔♠ sè ❝ë♥❣ t➼♥❤ g t❤ä❛ ♠➣♥✿ f (mx) m−→∞ m g(x) = lim ✭✷✳✶✷✮ 2m x ❱ỵ✐ ♠å✐ x ∈ R ✱ ❜✐➸✉ ❞✐➵♥ ❝❤✉♥❣ ♣❤ê ❜✐➳♥ ❧➔✿ g(x) = limm−→∞ f 2m ❚❤ü❝ t➳✱ ❍②❡rs ❝❤ù♥❣ ỵ ổ số ❣✐ú❛ ❝→❝ n ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ✣✐➲✉ tr➯♥ ✤÷đ❝ ①➙② ❞ü♥❣ ❝❤♦ ❜➜t ❝ù ❤➔♠ f : S −→ X ✱ ð ✤â X ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ S ❧➔ ♠ët ♥û❛ ♥❤â♠✳ ❑➳t q✉↔ ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✭tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t ❧➔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✮ ✈➔ ❜➔✐ t♦→♥ ➞♥ ❧✐➯♥ q✉❛♥ tỵ✐ ♥â trð t❤➔♥❤ ♣❤ê ❜✐➳♥ tr♦♥❣ ❝✉è✐ t❤➟♣ ♥✐➯♥✳ Ð ✤➙②✱ t❛ s➩ t➟♣ tr✉♥❣ ❞✉② ♥❤➜t ✈➔♦ sü ♠ð rë♥❣ ✤✐➲✉ ❦✐➺♥ q✉❡♥ t❤✉ë❝ ❝â t➼♥❤ ✤➲✉ ✤➦♥✭ ✈➼ ❞ư✱ f ❧➔ ✤♦ ✤÷đ❝ ❤♦➦❝ ❜à ❝❤➦♥ tr➯♥ tr➯♥ ♠ët t➟♣ ❞÷ì♥❣ ✤♦ ✤÷đ❝✮ ❤➔♠ ❝ë♥❣ t➼♥❤✱ ❦➳t ❤đ♣ g ❝â t❤➸ ❧✐➯♥ tư❝✳ ❚r♦♥❣ t➻♥❤ ❤✉è♥❣ ♥➔②✱ ❦➳t q✉↔ ❧✐➯♥ q✉❛♥ tỵ✐ ✤à♥❤ ♥❣❤➽❛ ✈➲ t➼♥❤ ê♥ ✤à♥❤ ❜❛♦ ❤➔♠ ✤✐➲✉ ❦✐➺♥ ✤÷đ❝ ♥â✐ ✤➳♥❀ ❜ð✐ ✈➻✱ ✤✐➲✉ ❦✐➺♥ ✤♦ ✤÷đ❝ ❦❤↔ t➼❝❤ ❧➔ ✤÷đ❝ ❜❛♦ ❤➔♠✳ ❚✉② ♥❤✐➯♥ ❝→❝ ❦➳t q✉↔ ✈➔ ❝→❝ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝ ♥â✐ ✤➳♥ ð ✤➙②✭ tr♦♥❣ tr÷í♥❣ ❤đ♣ ✤➦❝ ❜✐➺t✱ ✤✐➲✉ ❦✐➺♥ t➼♥❤ ê♥ ✤à♥❤ ❝õ❛ f ✮ ợ ởt tr ỳ ữ r ①↕ tø ❝→❝ ✤÷í♥❣ t❤➥♥❣ t❤ü❝ ❞÷ì♥❣ ✈➔♦ ❝❤➼♥❤ ♥â✮✳ ỵ S Rn ởt ♥û❛ ♥❤â♠ ♠➔ ♥â s✐♥❤ r❛ Rn ✈➔ ❝❤ù❛ ♠ët ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣ I✳ f : S −→ R t❤ä❛ tỗ t ởt (mk )k=1 ✱ s❛♦ ❝❤♦ ♠é✐ ❤➔♠ hk : S −→ C ◆➳✉ ❞➣② sè ✈æ ❤↕♥ ❝→❝ sè tü hk (x) := eif (mk x)/mk ✱ ✈ỵ✐ c ∈ Rn s❛♦ |f (x) c.x| ợ ữủ t tỗ t x S x ∈ S✳ ✤♦ ✤÷đ❝ tr➯♥ I✱ ❈❤ù♥❣ ♠✐♥❤✳ ◆❤÷ ✤➣ õ tữỡ tỹ tỗ t ởt t➼♥❤ g : S −→ R t❤ä❛ ♠➣♥ |f (x) − g(x)| ≤ ✱ ✈ỵ✐ ♠å✐ x ∈ S ✳ ❚ø ✭✷✳✶✷✮✱ t❛ ❝â✿ g(x) = lim k−→∞ f (mk x) mk ✈ỵ✐ ♠å✐ k ∈ N ✈➔ x ∈ S ✳ ◆❤í t➼♥❤ ❧✐➯♥ tư❝ ❝õ❛ ❤➔♠ ❧ơ② t❤ø❛✱ t❛ ❝â✿ h(x) := exp(ig(x)) = lim exp i k−→∞ f (mk x) mk = lim hk x k−→∞ ❱ỵ✐ ♠å✐ x ∈ S ✳ ❉♦ ✤â✱ sü ❤↕♥ ❝❤➳ ❝õ❛ h tỵ✐ I ❧➔ ♥ë✐ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ❤➔♠ ✤♦ ✤÷đ❝ ✈➔ ✈➻ ✈➟② ♥â ❧➔ ✤♦ ✤÷đ❝✳ ❱➻ g t❤ä❛ ♠➣♥ ✭✷✳✶✮ ✈➔ ✈➻ ♥û❛ ♥❤â♠ s✐♥❤ r❛ ♠ët ♥❤â♠ ❣✐❛♦ ❤♦→♥ ♠↕♥❤ s✐♥❤ r❛ ♥â ✭ ♥❤÷ t t tứ ỵ ợ A := S sỹ tỗ t c ∈ Rn ✱ s❛♦ ❝❤♦✿ g(x) = c.x✱ ✈ỵ✐ t➜t ❝↔ x ∈ S ✈➔ t❛ s✉② r❛ ✤✐➲✉ ♣❤↔✐ ❝❤ù♥❣ ♠✐♥❤✳ ◆❤➟♥ ①➨t ✷✳✻ ◆➳✉ f ❧➔ ✤ë ✤♦ ✤÷đ❝ ✤à❛ ♣❤÷ì♥❣ t❤➻ x −→ eif (mx)/x) ❧➔ ữỡ ợ m N hk tứ ỵ ✷✳✶✵ t❤ä❛ ♠➣♥ ✈➔ ✈➻ ✈➟② g ❧➔ ✤÷đ❝✳ ❍❛② ♥â✐ ❝→❝❤ ❦❤→❝✱ ♥➳✉ t❛ ❜✐➳t r➡♥❣ eif ❧➔ ✤♦ ✤÷đ❝ t❤➻ ♥â ❦❤ỉ♥❣ ✤õ ✤➸ s✉② r➡♥❣ g ❧➔ ✤♦ ✤÷đ❝✳ ❚❤➟t ✈➟②✱ ①➨t ❤➔♠ f := 2π[g(x)], x R tr ú ỵ õ g ♠ët ♥❣❤✐➺♠ ♣❤✐ t✉②➳♥ t➼♥❤ ❝õ❛ ✭✷✳✶✮ t❤➻ eif ❧➔ ❧✐➯♥ tö❝ ❧➫ ✈➻ ✤â ❧➔ ♠ët ❤➔♠ ❤➡♥❣ ✶✳ ✺✶ ❱➻ |t − [t]| ≤ ✈ỵ✐ ♠å✐ t ∈ R✱ t❛ ❝â✿ |f (x) − (2π)g(x)| ≤ 2π, ợ x R t ữỡ tr t ✤➥♥❣ t❤ù❝ t❛♠ ❣✐→❝ ✱ ✈➔ t➼♥❤ ❝ë♥❣ t➼♥❤ ❝õ❛ g ✱ s✉② r❛ r➡♥❣ f t❤ä❛ ♠➣♥ ✭✷✳✶✶✮✱ ✈ỵ✐ := 2g tọ ỵ ❍②❡rs s✉② r❛ r➡♥❣ ♥â ❧➔ ♠ët ❤➔♠ ❝ë♥❣ t➼♥❤ ♠➔ ✲ ①➜♣ ①➾ f ✱ ✈➔ ❝ô♥❣ s✉② r❛ r➡♥❣✿ f (mx) m−→∞ m ✈ỵ✐ ♠å✐ x ∈ R ợ õ t ữủ t t 2g(x) = lim ◆❤÷♥❣ g ❦❤ỉ♥❣ ✤♦ ✤÷đ❝ ✳ ❚❛ ❝â t❤➸ ♥â✐ r➡♥❣✿ ♠➦❝ ❞ị eif ❧➔ ✤♦ ✤÷đ❝ ♥❤÷♥❣ tứ ỵ r r tr ổ số ❤➔♠✿ gm (x) := eif (mx)/m , x ∈ R ❝❤➾ ❝â ❤ú✉ ❤↕♥ ❝❤ó♥❣ ❝â t❤➸ ❧➔ ✤♦ ✤÷đ❝✳ ▼ët sè ♥❤➟♥ ①➨t✿ ◆❤➟♥ ①➨t ✷✳✼ ✳ ❈→❝ ❤➺ ữỡ tr ỵ t q ✈➲ ❝→❝ ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ✭✷✳✶✮ ❝â t ữủ rở tợ ữỡ tr ✈➼ ❞ö✱ ❝❤♦ f : Rn −→ Rm t❤ä❛ ♠➣♥✭✷✳✶✮✱ ♥➳✉ f = (f1 , f2 , , fm )✈➔ eifk ữủ ợ ộ k {1, , m} t tỗ t ởt tr mìn s❛♦ ❝❤♦ f (x) = Cx ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ✣➙② ❧➔ ❦➳t q✉↔ ✤ì♥ ❣✐↔♥ ❝õ❛ ✣à♥❤ ỵ fk : Rn R tọ ♠➣♥ ✭✷✳✶✮✱ ✈ỵ✐ ♠é✐ k ✳ ◆❤➟♥ ①➨t ✷✳✽ ✳ ❑❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉✿ ▼ët ✤✐➲✉ t❤ó ✈à ❧➔ ❝â t❤➸ ♠ð rë♥❣ ❦➳t q✉↔ ♥➔② tỵ✐ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ✈ỉ ❤↕♥ ❝❤✐➲✉ tr➯♥ ✤â ①→❝ ✤à♥❤ ✤÷đ❝ ❤➔♠ sè ✤♦ ✤÷đ❝✳ ◆❤➟♥ ①➨t ✷✳✾ ✳ ❈→❝❤ ❝❤ù♥❣ ♠✐♥❤ ỵ sỷ q t tr ①➾ ❜❛♥ ✤➛✉ ✤÷đ❝ tr➻♥❤ ❜➔② ð ♣❤➛♥ ✷✳✶✳✸ ❣đ✐ þ r➡♥❣ ✤✐➲✉ ❦✐➺♥ ✈➲ q✉② t➢❝ ❝â t❤➸ ✤÷đ❝ ♠ð rë♥❣ ❤ì♥ ♥ú❛ ❜ð✐ q✉❛♥ ❤➺ ❦❤ỉ♥❣ ❣✐❛♥ ✤ë tr õ ởt tự ỵ tt trứ tữủ ỡ r ữợ t õ r➡♥❣ ♠ët ❜ë ❜❛ (A, B, F ) ❝â t➼♥❤ t➼❝❤ ♣❤➙♥ t✉➛♥ ❤♦➔♥ ♥➳✉ ♥❤ú♥❣ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷đ❝ t❤ä❛ ♠➣♥✿ ✺✷ ✶✳ A ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ sè t❤ü❝ ①→❝ ✤à♥❤ tr➯♥ Rn ❀ ✷✳ B ❧➔ t➟♣ ❤ñ♣ ❝→❝ ❤➔♠ sè ❝❤ù❛ {eif : g ∈ A}❀ ✸✳ F : B → C ❧➔ ♠ët ❤➔♠ sè✳ ✹✳ ❱ỵ✐ ♠å✐ β ∈ C t❤ä❛ ♠➣♥ |β| = 1✈➔ ♠å✐ h ∈ B ✱t❛ ❝â βh ∈ B ✈➔ F (βh) = βF (h);✳ ✺✳ ❍➔♠ sè x → c.x t❤✉ë❝ A ✈ỵ✐ t➜t ❝↔ c ∈ Rn A õ ữợ số ỳ t ữỡ ỗ t↕✐ ♠ët ❝ì sð {u1 , u2 , , un } ❝õ❛ Rn s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ g ∈ A t❤ä❛ ♠➣♥ q✉❛♥ ❤➺ g(x + uk ) = g(x) ✈ỵ✐ t➜t ❝↔ x ∈ Rn ✈➔ k ∈ {1, , n} ❍➔♠ sè gy ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐ gy (x) := g(x + y) ✈ỵ✐ ♠é✐ x ∈ Rn ❧➔ t❤✉ë❝ A ✈ỵ✐ ♠é✐ y ∈ Rn ✈➔ t❛ ❝â F (eigy ) = F (eig ); ✽✳ ❱ỵ✐ ộ g A tỗ t ởt số ỳ t✛ α > s❛♦ ❝❤♦ F (eiαg ) = ▼ët ✈➼ ❞ö ✈➲ ♠ët ❜ë ❜❛ (A, B, F ) ❧➔✿ A := {f : Rn −→ R : eif ❧➔ ✤ë ✤♦}, B := {eig : g ∈ A}, F (v) := I v(x)dx ✈ỵ✐ t➜t ❝↔ v ∈ A✱ ð ✤â I ⊂ Rn ❧➔ ♠ët ❤➻♥❤ ❧➟♣ ♣❤÷ì♥❣ ❜➜t ❦ý✳ ❇➙② ❣✐í ❣✐↔ sû r➡♥❣ t❛ ❝â ♠ët ❜ë ❜❛ (A, B, F ) t❤ä❛ ♠➣♥ t➼♥❤ t➼❝❤ ♣❤➙♥ t✉➛♥ ❤♦➔♥✳ ❚❛ ❦❤➥♥❣ ✤à♥❤ tr♦♥❣ tr÷í♥❣ ❤đ♣ ♥➔②✱ ♥➳✉ f ∈ A ✈➔ ♥➳✉ f tọ t tỗ t c Rn s❛♦ ❝❤♦ f (x) = c.x✱ ✈ỵ✐ ♠å✐ x ∈ Rn ✳ ❚❤➟t ✈➟②✱ ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝ơ♥❣ t÷ì♥❣ tü ♣❤➛♥ ❝❤ù♥❣ ♠✐♥❤ ❝õ❛ ♣❤➛♥ ✷✳✶✳✸✱ ð ✤➙② ❝â ❤❛✐ sü ❦❤→❝ ♥❤❛✉ ❝❤➼♥❤✿ ❚❤ù ♥❤➜t✱ tø ❇ê ✤➲ ✷✳✷ t❛ ❝➛♥ t❤❡♦ ❞✉② ♥❤➜t ❤❛✐ ✤♦↕♥ ✤➛✉✳ ❚❤ù ❤❛✐✱ s❛✉ ❦❤✐ →♣ ❞ö♥❣ ❤➔♠ t −→ eiαt , t ∈ R ✭ð ✤â α ❧➔ ♠ët sè ❤ú✉ t✛ ❞÷ì♥❣ tø t➼♥❤ ❝❤➜t (8) tr➯♥✮ ✱ t❛ →♣ ❞ư♥❣ ❤➔♠ sè F ❝❤♦ ❝↔ ❤❛✐ ✈➳ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤✿ eiαg(x+y) = eiαg(x) eiαg(y) , ✈➔ sû ❞ö♥❣ t➼♥❤ ❝❤➜t ✭✼✮ ð tr➯♥ tr♦♥❣ ✈✐➺❝ →♣ ❞ö♥❣ t➼❝❤ ♣❤➙♥ G(v) := I v(x)dx tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ✭ð ✤â I ❧➔ ♠ët s✐➯✉ ❤➻♥❤ ❤ë♣ s✐♥❤ r❛ ✺✸ ❜ð✐ ❝ì sð {u1 , , un } ✈➔ v ❧➔ ♠ët ❤➔♠ ❜➜t ❦ý ✈➔ ♥â ❧➔ t➼❝❤ ♣❤➙♥ tr➯♥ I ✮✱ ✈➔ sû ❞ư♥❣ ❇ê ✤➲ ✷✳✶ ♠ët ❝→❝❤ t÷ì♥❣ tü✳ ❚❤➟t t❤ó ✈à ❦❤✐ ❝â t❤➸ t➻♠ ✤÷đ❝ ♠ët ❜ë ❜❛ (A, B, F ) t❤ä❛ ♠➣♥ t➼♥❤ t✉➛♥ ❤♦➔♥ t➼❝❤ ♣❤➙♥ s❛♦ ❝❤♦ A = {f : Rn → R : eif ❧➔ ✤♦ ✤÷đ❝} ✈➔ F ❦❤ỉ♥❣ ♣❤↔✐ ♣❤➨♣ t♦→♥ t➼❝❤ ♣❤➙♥ ✭♥❣❤➽❛ ❧➔✱ F ❦❤ỉ♥❣ trị♥❣ ✈ỵ✐ v −→ I v(x)dx ❤♦➦❝ ♠ët sü ❜✐➳♥ ✤ê✐ ♥❤➭ ❝õ❛ ♣❤➨♣ t♦→♥ ♥➔② ✮ ❤♦➦❝ ❝❤♦ t❤➜② r➡♥❣ ✤➙② ❧➔ ✤✐➲✉ ❦❤æ♥❣ t❤➸✳ ✣✐➲✉ q✉❛♥ t➙♠ ❧➔ t➻♠ r❛ ➼t ♥❤➜t ♠ët ❜ë ❜❛ (A, B, F ) t❤ä❛ ♠➣♥ t➼♥❤ ❝❤➜t t➼❝❤ ♣❤➙♥ t✉➛♥ ❤♦➔♥ tr♦♥❣ sü ✤✐➲✉ ❝❤➾♥❤ ✈æ ❤↕♥ ❝❤✐➲✉ ❤♦➦❝ ❝❤➾ r❛ ❦❤æ♥❣ ❝â ởt tỗ t tỷ t F ❧➔ ♠ët ❤å ❝→❝ ❤➔♠ sè Fj : B −→ C✳ ❚❤ỉ♥❣ t❤÷í♥❣✱ ❜ð✐ ✈➻ ❝â ♥❤ú♥❣ ♥❣❤✐➺♠ ♣❤✐ t✉②➳♥ t➼♥❤ ❝õ❛ ✭✷✳✶✮✱ ♥❤ú♥❣ ♥❣❤✐➺♠ ♥➔② ❦❤æ♥❣ t❤✉ë❝ A✳ ❉♦ ✈➟②✱ ❜➜t ❦➸ A ✈➔ F ❧➔ ❣➻ ✱ A ❦❤ỉ♥❣ t❤➸ ❧➔ ♠ët t➟♣ ❤đ♣ t➜t ❝↔ ❝→❝ ❤➔♠ sè t❤ü❝✳ ▼ët sü ❜✐➳♥ ✤ê✐ t÷ì♥❣ ù♥❣ ❝õ❛ ♥❤ú♥❣ ❣➻ ✤÷đ❝ ✈✐➳t ❝→❝ ❝➙✉ ✈➔ ❝→❝ ✤♦↕♥ ð tr➯♥ ✤ó♥❣ ✈ỵ✐ ♠ët ❤➻♥❤ ①✉②➳♥ t♦♣♦✳ ❈✉è✐ ❝ị♥❣✱ ♥â s➩ r➜t ❝â ❣✐→ trà ✤➸ tê ❤ñ♣ ❝→❝ ✤✐➲✉ ❦✐➺♥ tữớ trứ tữủ ữủ tr ợ ởt ỵ tỗ t tữớ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✳ ✷✳✹ ▼ët sè ✈➼ ❞ư ♠✐♥❤ ❤å❛✳ ❱➼ ❞ö ✷✳✹✳✶✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f (x), g(x), h(x) ∈ ς(R) t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x + y) = g(x) + h(y), ∀x, y ∈ R ✭✷✳✶✸✮ ✭P❤÷ì♥❣ tr➻♥❤ ♥➔② ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r ♠ð rë♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✮✳ ▲í✐ ❣✐↔✐✿ ❚❤❛② y = ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮✱ t❛ ✤÷đ❝✿ g(x) = f (x) − h(0) = f (x) − b, b = h(0) ❚❤❛② x = ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✸✮ ✈➔ ✤➦t a = g(0)✱ t❛ ✤÷đ❝✿ h(y) = f (y) − a ❚❤❛② ❝→❝ ❦➳t q✉↔ tr➯♥ ✈➔♦ ✭✷✳✶✸✮✱ t❛ ❝â✿ f (x + y) = f (x) + f (y) − a − b, ∀x, y ∈ R ✺✹ ✣➦t F (x) = f (x) − a − b✱ ❦❤✐ ✤â✿ F (x) ∈ ς(R) ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ F (x + y) = F (x) + F (y), ∀x, y ∈ R ❚❤❡♦ ❦➳t q✉↔ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✱ t❛ ❝â✿ F (x) = cx ❙✉② r❛✿ f (x) = cx + a + b g(x) = cx + a ✈➔ h(x) = cx + b✳ ❱➟② f (x) = cx + a + b✱ g(x) = cx + a✱ h(x) = cx + b ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ✤➣ ❝❤♦✳ ❱➼ ❞ö ✷✳✹✳✷✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f (x), g(x), h(x) ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x + y) = g(x)h(y), ∀x, y ∈ R ✭✷✳✶✹✮ ▲í✐ ❣✐↔✐✿ ❚r♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ ❧➛♥ ❧÷đt t❤❛② x = 0, y = t ✈➔ y = 0, x = t t❛ ❝â✿ ◆➳✉ g(0) = 0, h(0) = 0✱ t❤➻✿    h(t) = f (t) , (a := g(0) = 0) f (t) = g(0).h(t) a ⇔ ✱ ∀t ∈ R✳ f (t)  f (t) = g(t).h(0)  g(t) = , (b := h(0) = 0) b ❚❤❛② ❝→❝ ❦➳t q✉↔ tr➯♥ ✈➔♦ ✭✷✳✶✹✮✱ t❛ ✤÷đ❝✿ f (x) f (y) a b f (x + y) f (x) f (y) ⇔ = , ∀x, y ∈ R ab ab ab f (x) ❚❛ ✤➦t Φ(x) = , x ∈ R✳ ab ❑❤✐ ✤â✿ Φ(x) ∈ ς(R) ✈➔ Φ(x + y) = Φ(x).Φ(y), ∀x, y ∈ R✳ ❙✉② r❛ Φ(x) = ecx , c ❧➔ ❤➡♥❣ sè✳ ❚ø ✤â✱ t❛ ❝â✿ f (x + y) = ✺✺  cx   f (x) = abe g(x) = aecx   h(x) = becx tr♦♥❣ ✤â✿a, b, c ❧➔ ❝→❝ ❤➡♥❣ sè✳ ❚❤û ❧↕✐ t❛ t❤➜② ❝→❝ ❤➔♠ sè tr➯♥ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮ ✤➣ ❝❤♦✳ ◆➳✉ a = 0✱ t❛ ❝â f (x) = 0, ∀x ∈ R✿ ✰ ◆➳✉ g ≡ t❤➻ h(x) ❧➔ ❤➔♠ sè tũ ỵ tỗ t x0 R s ❝❤♦ g(x0) = t❤➻ tr♦♥❣ ✭✷✳✶✹✮✱ ❝❤♦ x = x0✱ t❛ ✤÷đ❝✿ = f (x0 + y) = g(x0 )h(y), ∀y ∈ R ❙✉② r❛✿ h(y) = 0, ∀y ∈ R✳ ❚❤û ❧↕✐✱ t❛ t❤➜② ❝→❝ ❤➔♠ sè tr➯♥ t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✹✮✳ ◆➳✉ b = 0✱ t❛ ❝â f (x) = 0, ∀x ∈ R ✰ ◆➳✉ h ≡ t❤➻ g(x) ❧➔ ❤➔♠ sè tò② ỵ tỗ t x0 R s h(x0) = t❤➻ tr♦♥❣ ✭✷✳✶✹✮ ❝❤♦ x = x0✱ t❛ ✤÷đ❝✿ = f (x0 + y) = h(x0 )g(y), ∀y ∈ R ❙✉② r❛ g(y) = 0, ∀y ∈ R✳ ❚❤û ❧↕✐✱ t❛ t❤➜② ❝→❝ ❤➔♠ sè tr➯♥ tọ ữỡ tr ú ỵ r tr ♥❣❤✐➺♠ ð tr➯♥ ✤➣ ❜❛♦ ❤➔♠ ❝↔ ♥❣❤✐➺♠ t➛♠ t❤÷í♥❣ f ≡ 0, g ≡ 0, h ≡ ❚â♠ ❧↕✐✱ t❛ ❝â ❝→❝ ❦➳t q✉↔ s❛✉✿    cx  f (x) = abe  f ≡  f ≡0    ❤♦➦❝ g(x) = aecx ❤♦➦❝ g≡0 h≡0       h(x) = becx h(x) ∈ ς(R) g(x) ∈ ς(R) ✈ỵ✐ ♠å✐ x ∈ R✱ ✈➔ a, b, c ❧➔ ❝→❝ ❤➡♥❣ sè✳ ✺✻ ❑➌❚ ▲❯❾◆ ❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t→❝ ❣✐↔ ✤➣ tr➻♥❤ ❜➔② ✤÷đ❝ ♠ët sè ❦➳t q✉↔ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ❈ư t❤➸ ❧➔✿ ✶ ✳ ●✐ỵ✐ t❤✐➺✉ tê♥❣ q✉❛♥ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ r ự ỵ ❜ê ✤➲ ❧✐➯♥ q✉❛♥ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤②✳ ✸ ✳ ❚r➻♥❤ ❜➔② ✈➔ ❝❤ù♥❣ ♠✐♥❤ ♠ët sè ❜✐➳♥ t❤➸ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥❤÷✿ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥✱ ♣❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r✱✳✳✳ ✹ ✳ ❉ü❛ ✈➔♦ ❝→❝ ❦➳t q✉↔ ✤➣ tr➻♥❤ ❜➔② ð tr➯♥✱ ✈➟♥ ❞ö♥❣ ✈➔♦ ❣✐↔✐ ❝→❝ ❜➔✐ t♦→♥ ■❖▼ ✈➔ ❝→❝ ❜➔✐ t♦→♥ ❧✐➯♥ q✉❛♥✳ ❚r♦♥❣ t❤í✐ ❣✐❛♥ tỵ✐✱ t→❝ ❣✐↔ s➩ t✐➳♣ tư❝ t➻♠ ❦✐➳♠✱ ❤å❝ ❤ä✐ ✈➔ ♥❣❤✐➯♥ ❝ù✉ ✤➸ ❤♦➔♥ t❤✐➺♥ ❝→❝ ❦➳t q tr ỗ tớ tr ỗ t tự ✈ö ❝❤♦ q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❣✐↔♥❣ ❞↕② ❝õ❛ ♠➻♥❤ s❛✉ ♥➔②✳ ✺✼ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❚✐➳♥❣ ❱✐➺t ❬✶❪ ◆❣✉②➵♥ ❚➔✐ ❈❤✉♥❣✱ ▲➯ ❍♦➔♥❤ P❤á ✭✷✵✶✸✮✱ ❈❤✉②➯♥ ❦❤↔♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❬✷❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✷✵✶✹✮✱ ✤ê✐✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝ì ❜↔♥ ✈ỵ✐ ❜✐➳♥ sè ❜✐➳♥ ◆❳❇ ✣↕✐ ❤å❝ ◗✉è❝ ❣✐❛ ❍◆✳ ❬✸❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭❝❤õ ❜✐➯♥✮✱ ◆❣✉②➵♥ ❱➠♥ ❚✐➳♥ ✭✷✵✵✾✮✱ ởt số t ỗ ữù s ❣✐ä✐ tr✉♥❣ ❤å❝ ♣❤ê t❤ỉ♥❣✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❚✐➳♥❣ ❆♥❤ ❬✹❪ ❉❛♥✐❡❧ ❘❡❡♠ ✭✷✵✶✻✮✱ ❘❡♠❛r❦s ✈❛r✐❛t✐♦♥s ♦❢ ✐t✱ ❖♥ t❤❡ ❝❛✉❝❤② ❢✉♥❝t✐♦♥❛❧ ❡q✉❛t✐♦♥ ❛♥❞ ❛r❳✐✈✿✶✵✵✷✳✸✼✷✶✈✻✳ ❬✺❪ ❘❡❤❛❜ ❙❛❧❡❡♠ ❆❧✲♠♦s❛❞❞❡r ✭✷✵✶✷✮✱ ❖♥ st❛❜✐❧✐t② ♦❢ s♦♠❡ t②♣❡s ♦❢ ❢✉♥❝✲ t✐♦♥❛❧ ❡q✉❛t✐♦♥s✱ ▼❛st❡r ❚❤❡s✐s✱ ❙✉♣❡r✈✐s❡❞ ❜② ❉r✳ ❆✁s❛❞ ❨✳ ❆ ✬s❛❞✳ ... ♣❤÷ì♥❣ ♣❤→♣ ♣❤↔♥ ❝❤ù♥❣ ✤➸ ❝❤ù♥❣ ♠✐♥❤ ✤✐➲✉ ❞ü ✤♦→♥ ✤ó♥❣✳ ✽✳ ❚↕♦ ♥➯♥ ❝→❝ ❤➺ t❤ù❝ tr ỗ t t t số ứ ởt số ữợ tr t ❣✐↔ t➙♠ ✤➢❝ ♣❤➛♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ♥➯♥ ✤➣ ✤✐ s➙✉ ✈➔♦ ♥❣❤✐➯♥ ❝ù✉ ♥â✳ ✶✳✷ P❤÷ì♥❣... + y) = f (x) + f (y), ∀x, y ∈ R, ✤÷đ❝ t ỵ số ❧✐➯♥ tư❝ f (x) ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✮ ❦❤✐ ✈➔ ❝❤➾ ❦❤✐✿ f (x) = ax, ∀x ∈ R, tr õ a số tũ ỵ ❈❤ù♥❣ ♠✐♥❤✿ ❈❤♦ x = y ✱ t❛ ❝â✿ f (2x) =... x ∈ [m, n + 1] Am ❱➼ ❞ư ❦❤→❝✱ ❝â ❧➩ t❤ó ✈à ❤ì♥ ❧➔ f (x) := 2π[g(x)], x ∈ R ❦❤✐ [t] số ợ t ổ ữủt q số t❤ü❝ t ✈➔ g ❧➔ ♥❣❤✐➺♠ ❜➜t ❦➻ ❦❤æ♥❣ t✉②➳♥ t➼♥❤ ❝õ❛ ✭✷✳✶✮✳ ❚❤➟t ✈➟②✱ eif ≡ ♥❤÷♥❣ ❝â t❤➸