Bài giảng phương trình hàm (cho sinh viên ngành sư phạm toán)

✣❸■ ❍➴❈ ❙× P❍❸▼ ✣⑨ ◆➂◆● ❑❍❖❆ ❚❖⑩◆ ❇➔✐ ❣✐↔♥❣ Pì s ữ ❇✐➯♥ s♦↕♥✿ ❚❙✳ ❚ỉ♥ ❚❤➜t ❚ó ✣➔ ◆➤♥❣ ✲ ✽✴✷✵✷✶ ▼ư❝ ❧ư❝ ✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✶✳✶ ❑❤→✐ ♥✐➺♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✷ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✶ ❍➔♠ sè ❝❤➤♥✱ ❤➔♠ sè ❧➫ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✷ ❍➔♠ sè t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✸ ❍➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✸✳✹ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ✈➔ ♥❤➙♥ t➼♥❤ ✳ ✳ ✳ ✶✳✹ ❱➼ ❞ư ✈➲ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✶ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ f (x + a) = bf (x) + cx + d ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✷ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ F (f (x), f (α(x))) = ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✳✹✳✸ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ F (f (x), f (y), f (x + y), f (x − y), x, y) = ✶✳✺ ✣➦❝ tr÷♥❣ ❤➔♠ ❝õ❛ ♠ët sè ❤➔♠ sè ❝➜♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✶ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ Pữỡ tr ợ tỹ ❞♦ ✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✶ ❇➔✐ t♦→♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✷ ▼ët ✈➔✐ ❧í✐ ❣✐↔✐ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸ Pữỡ tr tr ợ ✳ ✳ ✳ ✳ ✷✳✶✳✸✳✶ ▲ỵ♣ ❤➔♠ ✤ì♥ ✤✐➺✉ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸✳✷ ▲ỵ♣ ❤➔♠ ❜à ❝❤➦♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✶✳✸✳✸ ▲ỵ♣ ❤➔♠ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝õ❛ ❤➔♠ ♠ô✱ ❤➔♠ ❧♦❣❛r✐t ✈➔ ❤➔♠ ❧ô② t❤ø❛ ✳ ✳ ✳ ✷✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝õ❛ ❤➔♠ ♠ơ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝õ❛ ❤➔♠ ❧♦❣❛r✐t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✷✳✸ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❝õ❛ ❤➔♠ ❧ô② t❤ø❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✸ P❤÷ì♥❣ tr➻♥❤ ❏❡♥s❡♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹ ❍➔♠ sè s✐♥❤ ❜ð✐ ❝→❝ ✤➦❝ tr÷♥❣ ❝õ❛ ❤➔♠ ❧÷đ♥❣ ❣✐→❝✱ ❤②♣❡r❜♦❧✐❝ ✈➔ ❧÷đ♥❣ ❣✐→❝ ♥❣÷đ❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✶ ❍➔♠ cos(ax) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✷ ❍➔♠ cosh(ax) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶ ✸ ✸ ✹ ✺ ✺ ✻ ✽ ✾ ✶✵ ✶✵ ✶✶ ✶✷ ✶✸ ✶✺ ✷✵ ✷✵ ✷✵ ✷✶ ✷✷ ✷✷ ✷✷ ✷✸ ✷✹ ✷✹ ✷✺ ✷✺ ✷✻ ✸✵ ✸✵ ✸✶ ✷ ▼Ö❈ ▲Ö❈ ✷✳✹✳✸ ❍➔♠ tan(ax) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✹ ❍➔♠ tanh(ax) ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✺ ❍➔♠ arcsin x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✹✳✻ ❍➔♠ arccos x ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✳✺ ❱➼ ❞ư ✈➲ ♣❤÷ì♥❣ tr ợ t ữỡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸ ❈→❝ ❞↕♥❣ tê♥❣ q✉→t ❝õ❛ P❚❍ ❈❛✉❝❤② ✈➔ ù♥❣ ❞ö♥❣ ✸✳✶ ▼ð rë♥❣ ❤➔♠ ❝ë♥❣ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷ ▼ët ✈➔✐ ❞↕♥❣ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✸✳✷✳✶ P❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✷✳✷ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣ f (ax + by + c) = pf (x) + qf (y) + r ✸✳✷✳✸ P❤÷ì♥❣ tr➻♥❤ ❈❛✉❝❤② ❝❤♦ ❤➔♠ ❤❛✐ ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✸ ❱➜♥ ✤➲ ✈➲ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹ Ù♥❣ ❞ö♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✶ ❚➼♥❤ ❞✐➺♥ t➼❝❤ ❤➻♥❤ ❝❤ú ♥❤➟t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✷ ❚➼♥❤ tê♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✸ ❚➻♠ ♣❤➙♥ ♣❤è✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✳✹✳✹ ❱➜♥ ✤➲ t✐➲♥ ❣û✐ ♥❣➙♥ ❤➔♥❣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✸ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✹✳✶ P❤÷ì♥❣ ♣❤→♣ ①➨t ❣✐→ trà ✳ ✳ ✳ ✳ ✹✳✷ P❤÷ì♥❣ ♣❤→♣ t❤➳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✸ ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❤➔♠ ❧✐➯♥ tö❝ ✹✳✹ ❙û ❞ö♥❣ ❞➣② sè ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✳✺ ❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ✤ì♥ →♥❤ ✳ ✳ ✳ ✹✳✻ ❙û ❞ư♥❣ ❝→❝ t➟♣ trị ♠➟t ✳ ✳ ✳ ✳ ✹✳✼ ❍➔♠ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✹ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✸✸ ✸✺ ✸✺ ✸✻ ✸✾ ✹✹ ✹✹ ✹✼ ✹✼ ✹✽ ✹✾ ✺✵ ✺✺ ✺✺ ✺✺ ✺✻ ✺✼ ✺✼ ✺✾ ✺✾ ✻✶ ✻✷ ✻✸ ✻✺ ✻✼ ✼✵ ✼✶ ❈❤÷ì♥❣ ✶ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✶✳✶ ❑❤→✐ ♥✐➺♠ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❝â ❞↕♥❣ A1 = A2 ✱ tr♦♥❣ ✤â A1 , A2 ❧➔ ♥❤ú♥❣ ❜✐➸✉ t❤ù❝ ❝❤ù❛ k ❜✐➳♥ ✤ë❝ ❧➟♣ x1 , , xk ✱ n ❤➔♠ ❝❤÷❛ ❜✐➳t F1 , , Fn ❝õ❛ j1 , , jn ❜✐➳♥ t÷ì♥❣ ù♥❣ ❝ơ♥❣ ♥❤÷ ♠ët sè ❤ú✉ ❤↕♥ ❝→❝ ❤➔♠ ✤➣ ❜✐➳t✳ ◆❤➟♥ ①➨t✿ ✐✮ ❑❤✐ ♥â✐ ✤➳♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ t❛ t❤÷í♥❣ ❧♦↕✐ trø r❛ ♣❤÷ì♥❣ tr➻♥❤ ✈✐ ♣❤➙♥✱ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✈➔ ❝→❝ ♣❤÷ì♥❣ tr➻♥❤ ❝❤ù❛ ♠ët sè ✈ỉ ❤↕♥ ❝→❝ ♣❤➨♣ t♦→♥✳ ✐✐✮ P❤➨♣ t♦→♥ ❝❤➼♥❤ tr♦♥❣ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♣❤➨♣ t♦→♥ t❤❛② t❤➳ ✭t❤❛② t❤➳ ❜✐➳♥ tr♦♥❣ ❝→❝ ❤➔♠ ✤➣ ❜✐➳t ❤♦➦❝ ❝❤÷❛ ❜✐➳t ❜ð✐ ❝→❝ ❤➔♠ ✤➣ ❜✐➳t ❤♦➦❝ ❝❤÷❛ ❜✐➳t✮✳ ❱➼ ❞ư ✶✳✶✳ ✲ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② f (x + y) = f (x) + f (y), ∀x, y ∈ R ✲ P❤÷ì♥❣ tr➻♥❤ P❡①✐❞❡r f (x + y) = g(x) + h(y), ∀x, y ∈ R ✲ P❤÷ì♥❣ tr➻♥❤ ❏❡♥s❡♥ f x+y = f (x) + f (y) , ∀x, y ∈ R ✲ P❤÷ì♥❣ tr➻♥❤ ❉✬❆❧❡♠❜❡rt f (x + y) + f (x − y) = 2f (x)f (y), ∀x, y ∈ R ✲ P❤÷ì♥❣ tr➻♥❤ ❝❤✉②➸♥ ✤ê✐ f (f (x, y), z) = f (x, g(y, z)), ∀x, y, z ∈ R ✸ ❈❍×❒◆● ✶✳ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✹ ✶✳✷ ◆❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✲ ◆❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ ♠ët ❤➔♠ sè ❤♦➦❝ ♠ët ❤➺ ❤➔♠ sè t❤ä❛ ♠➣♥ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ tr♦♥❣ ♠✐➲♥ ởt ợ F ữủ tờ qt ữỡ tr ợ ❤➔♠ ♥➔② ❧➔ t♦➔♥ ❜ë ❝→❝ ♥❣❤✐➺♠ r✐➯♥❣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤â✳ ✲ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❧➔ t➻♠ ♥❣❤✐➺♠ tê♥❣ q✉→t ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤â✳ ❱➼ ❞ư ✶✳✷✳ ❳➨t ❜➔✐ t♦→♥✿ ❚➻♠ ❤➔♠ f : R → R t❤♦↔ ✤✐➲✉ ❦✐➺♥✿ f (x + y) = f (x) + f (y), ∀x, y ∈ R ✲ ❍➔♠ f (x) = 4x, x ∈ R ❧➔ ♠ët ♥❣❤✐➺♠ r✐➯♥❣✳ ✲ ❍➔♠ f (x) = ax, x ∈ R ❧➔ ♥❣❤✐➺♠ tê♥❣ q✉→t✱ tr♦♥❣ ✤â a ❧➔ ♠ët ❤➡♥❣ số tỹ tũ ỵ f : R → R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ ❛✮ f (2x + 3) = 4(x + 1), ∀x ∈ R ❜✮ f (x − y) + 3f (x + y) = 4x + 2y, ∀x, y ∈ R t−3 ●✐↔✐✳ ❛✮ ✣➦t t = 2x + 3✱ s✉② r❛ x = ✳ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ t❛ ✤÷đ❝✿ t−3 + = 2(t − 1) f (t) = ❱➟②✱ f (x) = 2(x − 1), ∀x ∈ R✳ ❜✮ ❚ø ✤✐➲✉ ❦✐➺♥ s✉② r❛ f (x − y) + 3f (x + y) = 4x + 2y f (x + y) + 3f (x − y) = 4x − 2y ●✐↔✐ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ♥➔②✱ t❛ ✤÷đ❝ f (x + y) = x + y, ∀x, y ∈ R✳ ❚❤û ❧↕✐✱ f (x) = x ❧➔ ♥❣❤✐➺♠✳ ❱➟②✱ f (x) = x, ∀x ∈ R✳ ❄❄❄ ❈→❝❤ ❦❤→❝✿ ❱➼ ❞ö ✶✳✹✳ ❚➻♠ ❝→❝ ❤➔♠ f : R → R t❤ä❛✿ 2f (x + y) + 6y = f (x + 2y) + x3 , ∀x, y ∈ R ❱➼ ❞ö ✶✳✺✳ ❚➻♠ ❝→❝ ❤➔♠ f : [2; +∞) → R t❤ä❛✿ ❱➼ ❞ö ✶✳✻✳ 1 f (x + ) = x2 + , x > x x ❚➻♠ ❝→❝ ❤➔♠ f : R → R t❤♦↔ ✤✐➲✉ ❦✐➺♥✿ xf (y) − yf (x) = xy(x − y), ∀x, y ∈ R ❈❍×❒◆● ✶✳ ✺ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✶✳✸ ❚➼♥❤ ❝❤➜t ❝ì ❜↔♥ ❝õ❛ ❤➔♠ sè ✶✳✸✳✶ ❍➔♠ sè ❝❤➤♥✱ ❤➔♠ sè ❧➫ ✣à♥❤ ♥❣❤➽❛✿ ❈❤♦ ❤➔♠ sè f (x) ✈ỵ✐ t➟♣ ①→❝ ✤à♥❤ D(f ) ⊂ R✳ ✲ ❍➔♠ sè f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ sè ❝❤➤♥ tr➯♥ M, M ⊂ D(f ) ♥➳✉ ∀x ∈ M ⇒ −x ∈ M ✈➔ f (−x) = f (x) ✲ ❍➔♠ sè f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ sè ❧➫ tr➯♥ M, M ⊂ D(f ) ♥➳✉ ∀x ∈ M ⇒ −x ∈ M ✈➔ f (−x) = −f (x) ❱➼ ❞ö ✶✳✼✳ ✲ ❍➔♠ f (x) = sin(x) ❧➔ ❤➔♠ ❧➫ tr➯♥ R✳ ✲ ❍➔♠ f (x) = x2 ❧➔ tr [3, 3] t ỗ t ❤➔♠ ❝❤➤♥ ♥❤➟♥ trö❝ Oy ❧➔♠ trö❝ ✤è✐ ①ù♥❣✳ ✲ ỗ t t trử ❖①② ❧➔♠ t➙♠ ✤è✐ ①ù♥❣✳ ❱➼ ❞ö ✶✳✽✳ ❈❤ù♥❣ ♠✐♥❤ ❤➔♠ f (x) = ln 1−x 1+x ❧➔ ❤➔♠ ❝❤➤♥ tr➯♥ (−1; 1)✳ ❱➼ ❞ö ✶✳✾✳ ❈❤♦ ❤➔♠ sè f (x) = (m − 2)x2 + (m − 3)x + m2 − 4✳ ❛✮ ❚➻♠ m ✤➸ f (x) ❧➔ ❤➔♠ ❝❤➤♥ tr➯♥ R✳ ❜✮ ❚➻♠ m ✤➸ f (x) ❧➔ ❤➔♠ ❧➫ tr➯♥ R✳ ❱➼ ❞ö ✶✳✶✵✳ ▼é✐ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ R ✤➲✉ ❝â t❤➸ ♣❤➙♥ t➼❝❤ t❤➔♥❤ tê♥❣ ❝õ❛ ♠ët ❤➔♠ ❧➫ ✈➔ ♠ët ❤➔♠ ❝❤➤♥ tr➯♥ R✳ ●✐↔✐✳ ❳➨t ❤➔♠ sè f (x) ①→❝ ✤à♥❤ tr➯♥ R✳ ✣➦t 1 g(x) = (f (x) + f (−x)), h(x) = (f (x) − f (−x)) 2 ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ g(x) ❧➔ ❤➔♠ ❝❤➤♥ tr➯♥ R✱ h(x) ❧➔ ❤➔♠ ❧➫ tr➯♥ R ✈➔ f (x) = g(x) + h(x) ❈❍×❒◆● ✶✳ ✻ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❱➼ ❞ö ✶✳✶✶✳ ❈❤♦ x0 ∈ R✳ ❳→❝ ✤à♥❤ t➜t ❝↔ ❝→❝ ❤➔♠ sè f (x) s❛♦ ❝❤♦ f (x0 − x) = f (x), ∀x ∈ R x0 x0 ●✐↔✐✳ ✣➦t − x = t✳ ▲ó❝ ✤â x0 − x = + t ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❝â 2 ❞↕♥❣✿ x0 x0 f ( − t) = f ( + t), ∀t ∈ R 2 x0 x0 ✣➦t g(t) = f ( + t) ❤❛② f (t) = g(t − ) t❤➻ g(t) = g(−t), ∀t ∈ R ♥❣❤➽❛ 2 ❧➔ g ❧➔ ♠ët ❤➔♠ ❝❤➤♥ tr➯♥ R✳ x0 ❱➟②✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ f (t) = g(t − ) ✈ỵ✐ g ❧➔ ♠ët tũ ỵ tr R ❳→❝ ✤à♥❤ t➼♥❤ ❝❤➤♥ ❧➩ ❝õ❛ ❤➔♠ sè s❛✉ tr➯♥ t➟♣ ①→❝ ✤à♥❤✿ f (x) = |x + 2| + |x − 2| x2 − |x| ❱➼ ❞ö ✶✳✶✸✳ ❚➻♠ m ✤➸ ❤➔♠ sè s❛✉ ❧➔ ❤➔♠ sè ❝❤➤♥ tr➯♥ t➟♣ ①→❝ ✤à♥❤✿ f (x) = x2 (x2 − 2) + (2m2 − 2)x √ x2 + − m ự ỗ t số y = ax2 + bx + c, a ̸= ♥❤➟♥ ✤÷í♥❣ t❤➥♥❣ x = −b/(2a) ❧➔♠ trư❝ ✤è✐ ①ù♥❣✳ ❜✮ y = x3 − 3x2 + ♥❤➟♥ ✤✐➸♠ I(1, 0) ❧➔♠ t➙♠ ✤è✐ ①ù♥❣✳ ✶✳✸✳✷ ❍➔♠ sè t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✣à♥❤ ♥❣❤➽❛✿ ✲ ❍➔♠ sè f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ✭❝ë♥❣ t➼♥❤✮ ✈ỵ✐ ❝❤✉ ❦➻ a > tr➯♥ M ♥➳✉ M ⊂ D(f ) ✈➔ ∀x ∈ M ⇒ x ± a ∈ M f (x + a) = f (x), ∀x ∈ M ●✐↔ sû ❤➔♠ sè f (x) t✉➛♥ ợ T ố T ữủ ❧➔ ❝❤✉ ❦➻ ❝ì sð ♥➳✉ ♥❤÷ ❤➔♠ ❦❤ỉ♥❣ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❜➜t ❦➻ ❝❤✉ ❦➻ ♥➔♦ ❜➨ ❤ì♥ T ✳ ✲ ❍➔♠ sè f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✭❝ë♥❣ t➼♥❤✮ ✈ỵ✐ ❝❤✉ ❦➻ a > tr➯♥ M ♥➳✉ M ⊂ D(f ) ✈➔ ∀x ∈ M ⇒ x ± a ∈ M f (x + a) = −f (x), ∀x ∈ M ❈❍×❒◆● ✶✳ ✼ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ●✐↔ sû f (x) ❧➔ ❤➔♠ sè ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ T ✳ ❙è T ✤÷đ❝ ❣å✐ ❧➔ ❝❤✉ ❦➻ ❝ì sð ♥➳✉ ♥❤÷ ❤➔♠ f (x) ❦❤ỉ♥❣ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❜➜t ❦➻ ❝❤✉ ❦➻ ♥➔♦ ❜➨ ❤ì♥ T ✳ ❱➼ ❞ö ✶✳✶✺✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ❤➔♠ sè f (x) = sin x t✉➛♥ ❤♦➔♥ tr➯♥ R ✈ỵ✐ ❝❤✉ ❦➻ ❝ì s T = sỷ tỗ t T ∈ (0, 2π) s❛♦ ❝❤♦ ❤➔♠ sin x t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ T ✳ ▲ó❝ ✤â sin x = sin(T +x), ∀x ∈ R✳ ❈❤♦ x = t❤➻ sin T = 0✳ ❙✉② r❛ T = π ✳ ✣✐➲✉ ♥➔② ✈æ ❧➼✳ ❱➟② ❤➔♠ sin x t✉➛♥ ❤♦➔♥ ✈ỵ✐ ❝❤✉ ❦➻ ❝ì sð T = 2π ✳ ❱➼ ❞ö ✶✳✶✻✳ ❳➨t ❤➔♠ ❉✐r✐❝❤❧❡t ①→❝ ✤à♥❤ t❤❡♦ ❤➺ t❤ù❝ f (x) = 0, x ∈ Q 1, x ∈ /Q ❉➵ ❞➔♥❣ t❤➜② r➡♥❣ ♠å✐ sè ❤ú✉ t➾ ❞÷ì♥❣ ✤➲✉ ❧➔ ❝❤✉ ❦➻ t✉➛♥ ❤♦➔♥ ❝õ❛ ❤➔♠ f (x) ✈➔ ✤✐➲✉ ♥➔② ❞➝♥ ✤➳♥ ❤➺ q✉↔✿ ❤➔♠ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤÷♥❣ ❦❤ỉ♥❣ ❝â ❝❤✉ ❦➻ ❝ì sð✳ ❱➼ ❞ư ✶✳✶✼✳ ❈❤ù♥❣ ♠✐♥❤ ❤➔♠ cos(x2) ❦❤ỉ♥❣ ♣❤↔✐ ❤➔♠ ❤➔♠ t✉➛♥ ❤♦➔♥ tr➯♥ R✳ ●✐↔✐✳ ●✐↔ sû tỗ t T > s cos (x + T )2 = cos(x2 ), ∀x ∈ R✳ √ ❈❤♦ x = t❛ ✤÷đ❝ cos T = ⇒ T = 2kπ, k ∈ N∗ ✳ √ √ √ 2 ❈❤♦ x = π t❛ ✤÷đ❝ cos ( π + T ) = −1 ⇒ ( π + T ) = (2n + 1)π, n ∈ N✳ √ √ ❙✉② r❛✿ + 2k = 2n + ⇔ 2k = 2n − 2k ∈ N √ π ❈❤♦ x = t❛ ✤÷đ❝ cos ( π + T ) = ❤❛② π √ π ( + 2kπ)2 = (2m + 1) , m ∈ N 2 √ √ ❙✉② r❛ (1 + k)2 = 2m + ❤❛② k = m − 2k ✳ ❚ø ✤â✿ √ √ 2k 2n − 2k 2= √ = ∈Q m − 2k k ✣✐➲✉ ♥➔② ✈ỉ ❧➼✳ ❈❍×❒◆● ✶✳ ✽ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❱➼ ❞ư ✶✳✶✽✳ ❳➨t t➼♥❤ t✉➛♥ ❤♦➔♥ ✈➔ t➻♠ ❝❤✉ ❦➻ ❝ì sð ✭♥➳✉ ❝â✮✿ ❛✮ f (x) = cos x2 cos 3x √ ❜✮ f (x) = cos x + cos 2x ❱➼ ❞ö ✶✳✶✾✳ ❈❤♦ a, b, c, d ̸= 0✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ f (x) = a sin cx + b cos dx ❧➔ ❤➔♠ sè t✉➛♥ ❤♦➔♥ tr➯♥ R ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ c/d ❧➔ sè ❤ú✉ t➾✳ ❱➼ ❞ö ✶✳✷✵✳ ❈❤ù♥❣ ♠✐♥❤ ❤➔♠ sin(x2) ❦❤æ♥❣ ♣❤↔✐ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ tr➯♥ R ❱➼ ❞ö ✶✳✷✶✳ ❈❤♦ ❤➔♠ f (x) ①→❝ ✤à♥❤ tr➯♥ R ✈➔ t❤ä❛✿ f (x + 1) = + f (x) − f (x)2 ❈❤ù♥❣ ♠✐♥❤ f (x) ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ tr➯♥ R✳ ✶✳✸✳✸ ❍➔♠ t✉➛♥ ❤♦➔♥ ✈➔ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ✣à♥❤ ♥❣❤➽❛✿ ✲ ❍➔♠ f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦➻ a ✭a ∈ / {−1, 0, 1}✮ tr➯♥ ♥➳✉ M ∈ D(f ) ✈➔ ▼ ∀x ∈ M ⇒ a±1 x ∈ M f (ax) = f (x), ∀x ∈ M ✲ ❍➔♠ f (x) ✤÷đ❝ ❣å✐ ❧➔ ❤➔♠ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦➻ a ✭a ∈ / {−1, 0, 1}✮ tr➯♥ ♥➳✉ M ∈ D(f ) ✈➔ ▼ ∀x ∈ M ⇒ a±1 x ∈ M f (ax) = −f (x), ∀x ∈ M ❱➼ ❞ö ✶✳✷✷✳ ❍➔♠ f (x) = cos (2π log3x)✱ g(x) = sin (π log3x) ❧➔ ♠ët ❤➔♠ t✉➛♥ ❤♦➔♥✱ ♣❤↔♥ t✉➛♥ ❤♦➔♥ ♥❤➙♥ t➼♥❤ ❝❤✉ ❦➻ ✸ tr➯♥ R+ = (0; +∞)✳ ❱➼ ❞ö ✶✳✷✸✳ ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ ♥➳✉ ❤➔♠ f (x) ①→❝ ✤à♥❤ tr➯♥ R✱ ❧✐➯♥ tö❝ t↕✐ x = ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ∃a ̸= 0, |a| = ̸ 1, f (ax) = f (x), ∀x ∈ R t❤➻ f (x) ≡ const, ∀x ∈ R✳ ●✐↔✐✳ ❛✮ ❚r÷í♥❣ ❤đ♣ |a| < 1✳ ▲ó❝ ✤â ∀x ∈ R t❛ ❝â f (x) = f (ax) = = f (an x), ∀n ∈ N ❈❍×❒◆● ✶✳ ✾ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❉♦ ❤➔♠ f ❧✐➯♥ tư❝ t↕✐ x = ♥➯♥ f (x) = lim f (an x) = f ( lim an x) = f (0) n→+∞ n→+∞ ❜✮ ❚r÷í♥❣ ❤đ♣ |a| > 1✳ ✣✐➲✉ ❦✐➺♥ ❜➔✐ t♦→♥ ✤÷đ❝ ✈✐➳t ❧↕✐ f (x) = f x , ∀x ∈ R a ⑩♣ ❞ö♥❣ ❦➳t q✉↔ ð ♣❤➛♥ ❛✮ ❝❤♦ tr÷í♥❣ ❤đ♣ b = 1/a✱ t❛ ❝ơ♥❣ ✤÷đ❝ f (x) = f (0), ∀x ∈ R✳ ❱➟②✱ f (x) ≡ const, ∀x ∈ R✳ ✶✳✸✳✹ ▼è✐ ❧✐➯♥ ❤➺ ❣✐ú❛ ❤➔♠ t✉➛♥ ❤♦➔♥ ❝ë♥❣ t➼♥❤ ✈➔ ♥❤➙♥ t➼♥❤ ❇➔✐ t♦→♥✿ ❈❤♦ a ∈/ {−1, 0, 1}✳ ❳→❝ ✤à♥❤ ❝→❝ ❤➔♠ f (x) s❛♦ ❝❤♦ f (ax) = f (x), ∀x ∈ R ●✐↔✐✳ ❳➨t ✷ tr÷í♥❣ ❤đ♣✿ ❛✮ ❱ỵ✐ a > 0✿ ✲ ◆➳✉ x > t❤➻ ✤➦t x = at ✈➔ f (at ) = h1 (t)✳ ❑❤✐ ✤â t❛ ✤÷đ❝ t = loga x ✈➔ h1 (t + 1) = h1 (t), ∀t ∈ R ✲ ◆➳✉ x < t❤➻ ✤➦t −x = at ✈➔ f (−at ) = h2 (t)✳ ❑❤✐ ✤â t❛ ✤÷đ❝ t = loga |x| ✈➔ h2 (t + 1) = h2 (t), ∀t ∈ R ❜✮ ❱ỵ✐ a < 0✿ ❑❤✐ ✤â f (a2 x) = f (x) ✈➔ ữỡ tr ữủ ữợ ❞↕♥❣ f (x) = (g(x) + g(ax)) tr♦♥❣ ✤â g(a2 x) = g(x)✳ ❚❤➟t ✈➟②✱ ♥➳✉ f ❝â ❜✐➸✉ ❞✐➵♥ ð ❞↕♥❣ tr➯♥ t❤➻ f (ax) = f (x)✳ ◆❣÷đ❝ ❧↕✐✱ ♥➳✉ f (ax) = f (x) t❤➻ f (x) = (f (x) + f (ax)) ❚ø ❜✐➸✉ ❞✐➵♥ tr➯♥✱ t❛ t❤✉ ✤÷đ❝ ♥❣❤✐➺♠ ❜➡♥❣ ❝→❝❤ →♣ ❞ư♥❣ ❦➳t q✉↔ ð tr÷í♥❣ ❤đ♣ ❛✮✳ ❈❤÷ì♥❣ ✹ ❈→❝ ♣❤÷ì♥❣ ♣❤→♣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✹✳✶ P❤÷ì♥❣ ♣❤→♣ ①➨t ❣✐→ trà ❈❤♦ ❝→❝ ❜✐➳♥ ♥❤➟♥ ❝→❝ ❣✐→ trà ✤➦❝ ❜✐➺t ✤➸ t↕♦ r❛ ♥❤✐➲✉ ✤✐➲✉ ❦✐➺♥ ❝❤♦ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✤❛♥❣ ①❡♠ ①➨t✳ ❱➼ ❞ö ✹✳✶✳ ❚➻♠ ❤➔♠ f : R+ → R+ t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (1) = 1/2; f (xy) = f (x)f (3/y) + f (y)f (3/x), ∀x, y > ●✐↔✐✳ ❈❤♦ x = 1; y = t❛ ✤÷đ❝ f (3) = 1/2✳ ❈❤♦ x = 1✱ f (y) = f (3/y)✳ ▲ó❝ ✤â✱ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉ trð t❤➔♥❤✿ f (xy) = 2f (x)f (y), ∀x, y > ❚❤❛② y = 3/x✱ t❛ ✤÷đ❝✿ f (3) = 2f (x)f (3/x) ⇔ 1/2 = 2(f (x))2 ⇔ f (x) = 1/2 ❚❤û ❧↕✐ ✤ó♥❣✳ ❱➟②✱ f (x) = 1/2, ∀x > 0✳ ❱➼ ❞ö ✹✳✷✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x) ≥ ✈➔ f (x + y) ≥ f (x) + f (y), ∀x, y ∈ R ●✐↔✐✳ ❈❤♦ x = y = 0✱ t❛ ✤÷đ❝✿ f (0) ≥ ⇒ f (0) = f (0) ≥ 2f (0) ❈❤♦ y = −x✱ t❛ ❝â ≥ f (x) + f (−x) f (x) ≥ 0, f (−x) ≥ ⇒ f (x) = f (−x) = 0, ∀x ❚❤û ❧↕✐ ✤ó♥❣✳ ❱➟②✱ f (x) = 0, ∀x ∈ R✳ ✺✾ ❈❍×❒◆● ✹✳ ✻✵ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❱➼ ❞ư ✹✳✸✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (0) = a; f (π/2) = b (a, b = const) f (x + y) + f (x − y) = 2f (x) cos y, ∀x, y ∈ R ●✐↔✐✳ ❈❤♦ y = π/2✱ t❛ ✤÷đ❝ f (x + π/2) + f (x − π/2) = 0✳ ❈❤♦ x = 0✿ f (y) + f (−y) = 2a cos y ❈❤♦ x = π/2✿ f (π/2 + y) + f (π/2 − y) = 2b cos y ❚❛ ✤÷đ❝ ❤➺✿    f (x + π/2) + f (x − π/2) = f (x + π/2) + f (π/2 − x) = 2a cos(x − π/2) = 2a sin x   f (x + π/2) + f (π/2 − x) = 2b cos x ⇒ f (x) = a cos x+b si ❚❤û ❧↕✐ ✤ó♥❣✳ ❱➟②✱ f (x) = a cos x + b sin x, ∀x ∈ R✳ ❱➼ ❞ö ✹✳✹✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x)f (y) = f (x + y) + sin x sin y, ∀a, y ∈ R ●✐↔✐✳ ❈❤♦ x = y = 0✿ f (0)(1 − f (0)) = ⇔ f (0) = ❤♦➦❝ f (0) = 1✳ ❚r÷í♥❣ ❤đ♣ f (0) = 0✿ ❈❤♦ y = 0✿ f (x) = 0✱ ✤➙② ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ♥❣❤✐➺♠✳ ❚r÷í♥❣ ❤đ♣ f (0) = 1✿ ❈❤♦ y = −x✿ f (x)f (−x) = − sin2 x = cos2 x✳ ❈❤♦ x = π/2✿ f (π/2)f (−π/2) = ⇔ f (π/2) = ❤♦➦❝ f (−π/2) = 0✳ ✲ ◆➳✉ f (π/2) = 0✿ ❈❤♦ y = π/2✱ f (x + π/2) + sin x = ⇒ f (x) = cos x ✲ ◆➳✉ f (−π/2) = 0✿ ❈❤♦ y = −π/2✱ f (x − π/2) − sin x = ⇒ f (x) = cos x ❚❤û ❧↕✐✱ f (x) = cos x ❧➔ ♥❣❤✐➺♠✳ ❱➟②✱ f (x) = cos x, ∀x ∈ R✳ ❱➼ ❞ö ✹✳✺✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x + y) + f (x − y) − 2f (x)f (y + 1) = 2xy(3y − x2 ), ∀x, y ∈ R ❱➼ ❞ö ✹✳✻✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x + y) ≥ f (x)f (y) ≥ 2020x+y , ∀x, y ∈ R ❱➼ ❞ö ✹✳✼✳ ❚➻♠ ❤➔♠ f : [a, b] → [a, b] t❤ä❛✿ |f (x) − f (y)| ≥ |x − y|, ∀x, y ∈ [a, b] ❱➼ ❞ö ✹✳✽✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x)f (y) = f (x − y) − sin x sin y, ∀a, y ∈ R ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● PP Pì Pữỡ t ❚❤❛② t❤➳ x = φ(t) ❤♦➦❝ t = φ(x) ✤➸ t r ữỡ tr ợ ữỡ tr➻♥❤ ❤➔♠✳ ❱➼ ❞ö ✹✳✾✳ ❚➻♠ ❤➔♠ f : R\{2} → R t❤ä❛✿ f 2x + x−1 = x2 + 2x, ∀x ̸= ●✐↔✐✳ ✣➦t t = 2xx −+11 ✱ t❛ ✤÷đ❝ x = tt −+ 12 ✳ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉✿ 3t2 − , ∀t ̸= f (t) = (t − 2)2 3x2 − ❚❤û ❧↕✐ ✤ó♥❣✳ ❱➟②✱ f (x) = , ∀x ̸= 2✳ (x − 2)2 ❱➼ ❞ö ✹✳✶✵✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x) + xf (−x) = x + 1, ∀x ∈ R ●✐↔✐✳ ❚❤❛② x ❜ð✐ −x✱ t❛ ✤÷đ❝ ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿ f (x) + xf (−x) = x + −xf (x) + f (−x) = −x + ●✐↔✐ r❛✱ f (x) = 1✳ ❚❤û ❧↕✐ ✤ó♥❣✳ ❱➟②✱ f (x) = 1, ∀x ∈ R✳ ❱➼ ❞ö ✹✳✶✶✳ ❚➻♠ ❤➔♠ f : R\{1} → R t❤ä❛✿ f (1 + ) = x2 + 1, ∀x ̸= x ❱➼ ❞ö ✹✳✶✷✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛✿ ❛✮ 2f (x) + f (1 − x) = x2 , ∀x ∈ R ❜✮ (x − y)f (x + y) − (x + y)f (x − y) = 4xy(x2 − y ), ∀x, y ∈ R ❈❍×❒◆● ✹✳ ✻✷ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✹✳✸ ❙û ❞ö♥❣ t➼♥❤ ❝❤➜t ❤➔♠ ❧✐➯♥ tö❝ ❚❛ ❝â✿ f (x) ❧✐➯♥ tö❝ t↕✐ x0 ⇔ lim f (x) = f (x0 ) x→x0 ⇔ ∀ {xn } , xn → x0 t❤➻ lim f (xn ) = f (x0 ) ❤❛② lim f (xn ) = f ( lim xn ) n→∞ n→∞ n→∞ ❱➼ ❞ö ✹✳✶✸✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x) = f (3x), ∀x ∈ R ●✐↔✐✳ ✣✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦ ✤÷đ❝ ✈✐➳t ❧↕✐✿ f (x) = f x , ∀x ∈ R✳ x , ∀x ∈ R, ∀n ≥ 1✳ 3n ❉♦ t➼♥❤ ❧✐➯♥ tö❝ ❝õ❛ ❤➔♠ f ♥➯♥ ✈ỵ✐ ① ❝è ✤à♥❤✱ t❛ ❝â x x f (x) = lim f n = f lim n = f (0) n→+∞ n→+∞ 3 ❚ø ✤â t❛ ✤÷đ❝ f (x) = f ❱➟② f (x) ≡ const ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❱➼ ❞ö ✹✳✶✹✳ ❈❤♦ α ∈ R, α ̸= ±1✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f (x) ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R+ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (xα ) = f (x), ∀x ∈ R+ ●✐↔✐✳ ❛✮ ❚r÷í♥❣ ❤đ♣ |α| < 1✿ ❚❛ ✤÷đ❝ n f (x) = f (xα ), ∀n ∈ N ❙✉② r❛ n f (x) = lim f (xα ) = f (1), ∀x ∈ R+ n→+∞ ❜✮ ❚r÷í♥❣ ❤đ♣ |α| > 1✿ ✣✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦ ✤÷đ❝ ✈✐➳t ❧↕✐ f (x) = f (x α ), ∀x ∈ R+ ❚÷ì♥❣ tü tr÷í♥❣ ❤đ♣ ✶✱ t❛ ❝ơ♥❣ ✤÷đ❝ f (x) = f (1), ∀x ∈ R+ ❱➟② f (x) ≡ c ∈ R ❜➜t ❦➻✱ ∀x ∈ R+ ✳ ❱➼ ❞ö ✹✳✶✺✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (4x) + f (25x) = 2f (10x), ∀x ∈ R ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✻✸ ●✐↔✐✳ ✣✐➲✉ ❦✐➺♥ ✤➣ ❝❤♦ ✤÷đ❝ ✈✐➳t ❧↕✐ f ( x) + f ( x) = 2f (x), ∀x ∈ R ❤❛② f ( x) − f (x) = f (x) − f ( x), ∀x ∈ R 2 ✣➦t g(x) = f ( x) − f (x)✱ t❛ ✤÷đ❝ g(0) = ✈➔ g(x) = g x , ∀x ∈ R 2 ❚÷ì♥❣ tü✱ s✉② r❛ g(x) ≡ 0, ∀x ∈ R✳ ❉♦ ✤â f ( x) = f (x), ∀x ∈ R✳ ❱➟② f (x) ≡ const, ∀x ∈ R✳ ❱➼ ❞ö ✹✳✶✻✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ f (2x) + f (5x) = 0, ∀x ∈ R ❱➼ ❞ö ✹✳✶✼✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tö❝ tr➯♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ ❛✮ f (x) = f (2x) + 5x, ∀x ∈ R ❜✮ f (x) = f (x2 ) + x(x − 1), ∀x ∈ R ❝✮ f (x2 ) + f (x) = x2 + x, ∀x ∈ R ❞✮ 3f (2x) = f (x) + 5x, ∀x ∈ R ❱➼ ❞ö ✹✳✶✽✳ ❈❤♦ t ∈ (0, 1)✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ sè f ❧✐➯♥ tö❝ tr➯♥ R ✈➔ t❤♦↔ ✤✐➲✉ ❦✐➺♥✿ f (x) − 2f (tx) + f (t2 x) = x2 , ∀x ∈ R ✹✳✹ ❙û ❞ö♥❣ số f (x) t ữợ ❞↕♥❣ ❞➣② ❤➔♠ f (x) = gn (x), n ∈ N s tỗ t ợ lim gn (x) ✈ỵ✐ ♠é✐ x ❝è ✤à♥❤✳ n→∞ ❱➼ ❞ư ✹✳✶✾✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f : R → R✱ ❜à ❝❤➦♥ tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ x = ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ 2f (2x) = f (x) + x, ∀x ∈ R ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✻✹ ●✐↔✐✳ ✣✐➲✉ ❦✐➺♥ ❜➔✐ t♦→♥ ✤÷đ❝ ✈✐➳t ❧↕✐✿ x x 1 x x x x x x + = f + + = f + + = = f (x) = f 2 2 4 4 16 x x x x = nf n + + + + n , ∀n ∈ N 2 16 x f (x) = lim n f n + lim n→+∞ n→+∞ 2 ❉➵ t❤➜② r➡♥❣ f (x) = n k=1 x x = 4k x ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✳ ❱➼ ❞ö ✹✳✷✵✳ ❚➻♠ t➜t ❝↔ ❝→❝ ❤➔♠ f : R → R✱ ❜à ❝❤➦♥ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ ✤✐➸♠ x = ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ x f (x) − f = x − x2 , ∀x ∈ R 2 ●✐↔✐✳ ❈❤♦ x = t❛ ✤÷đ❝ f (0) = 0✳ ▼➦t ❦❤→❝✱ t❤❛② x ❧➛♥ ❧÷đt ❜ð✐ x/2, x/4, x/8, t❛ ✤÷đ❝ x = x − x2 f (x) − f 2 x x x 1 x f − f = − = 2 4 2 x x x 1 x f − f = − = 4 8 4 x x x x2 f n − n+1 f n+1 = n − n 2n 2 x x2 − x2 x − 42 83 ❉♦ ✤â f (x)− x 1 1 1 f = x + + + + −x + + + + 2n+1 2n+1 42 4n 82 8n ❈❤♦ n → +∞ t❛ ✤÷đ❝ f (x) = x − x2 , ∀x ∈ R ❱➼ ❞ö ✹✳✷✶✳ ❚➻♠ ❝→❝ ❤➔♠ f : R → R t❤ä❛ ♠➣♥✿ f (f (x)) = 3f (x) − 2x, ∀x ∈ R ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✻✺ ●✐↔✐✳ ❱ỵ✐ x ∈ R ❝è ✤à♥❤✱ ✤➦t u0 = x, u1 = f (x), un+1 = f (un), n ≥ 1✳ ❚❛ ✤÷đ❝ ❞➣② (un ) t❤ä❛✿ un+2 = 3un+1 − 2un , n ≥ P❤÷ì♥❣ tr➻♥❤ ✤➦❝ tr÷♥❣✿ λ2 − 3λ + = ❝â ♥❣❤✐➺♠ λ1 = 1, λ2 = 2✳ ❉♦ ✤â✿ un = c1 + c2 2n ✳ ❚ø ✤â✿ u0 = c1 + c2 = x u1 = c1 + 2c2 = f (x) ⇔ c1 = 2x − f (x) c2 = f (x) − x ❱➟②✱ f (x) = x + c ❤♦➦❝ f (x) = 2x + c✳ ❱➼ ❞ö ✹✳✷✷✳ ❚➻♠ ❤➔♠ f : R → R ❧✐➯♥ tö❝ t↕✐ x = ✈➔ t❤ä❛✿ f (3x) + f (x/3) = 10x , ∀x ∈ R ❱➼ ❞ö ✹✳✷✸✳ ❈❤♦ a, b > 0✳ ❚➻♠ ❤➔♠ f : [0, +∞) → [0, +∞) t❤ä❛✿ f (f (x)) + a.f (x) = b(a + b)x, ∀x ≥ ✹✳✺ ❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ✤ì♥ →♥❤ ●✐↔ sû ❤➔♠ f ❝➛♥ t➻♠ ✤ì♥ ✤✐➺✉ ✭♥❣➦t✮ ❤❛② ✤ì♥ →♥❤ tr➯♥ ♠✐➲♥ ①→❝ ✤à♥❤✳ ▲ó❝ ✤â tø ✤➥♥❣ t❤ù❝ f (A) = f (B) s✉② r❛ A = B ✳ ❱➼ ❞ö ✹✳✷✹✳ ❚➻♠ ❝→❝ ❤➔♠ f (x) ①→❝ ✤à♥❤✱ ♥❣❤à❝❤ ❜✐➳♥ tr♦♥❣ (0, +∞) ✈➔ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x + f (y)) = ●✐↔✐✳ ❚❛ ❝â f y−1 + f (y) y = y , ∀x, y ∈ R+ xy + y = 1, ∀y ∈ R+ y−1 y+1 y ❚ø ✤â s✉② r❛ f y−1 + f (y) y =f x−1 + f (x) , ∀x, y ∈ R+ x ❉♦ ❤➔♠ f ♥❣❤à❝❤ ❜✐➳♥ ♥➯♥ y−1 x−1 + f (y) = + f (x), ∀x, y ∈ R+ y x ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ❤❛② +c x ❚❤û ❧↕✐ ✤✐➲✉ ❦✐➺♥ t❛ ✤÷đ❝ c = 0✳ ❱➟②✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥ ❧➔ f (x) = f (x) = , ∀x ∈ R+ x ❱➼ ❞ö ✹✳✷✺✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛✿ f (f (x) + 2y) = 4x + 4y + 4, ∀x, y ∈ R ●✐↔✐✳ ❉➵ t❤➜②✱ f ✤ì♥ →♥❤✳ ▼➦t ❦❤→❝✱ f (f (x) + 2y) = 4x + 4y + = f (f (y) + 2x), ∀x, y ∈ R ❙✉② r❛✿ f (x) + 2y = f (y) + 2x, ∀x, y ∈ R ❤❛② f (x) − 2x = f (y) − 2y, ∀x, y ∈ R ❉♦ ✤â✿ f (x) − 2x = c, ∀x ∈ R✳ ❚❤û ❧↕✐✱ f (x) = 2x + c ❧➔ ♥❣❤✐➺♠✳ ❱➟②✱ f (x) = 2x + c, x ∈ R ✈ỵ✐ c = const✳ ❱➼ ❞ư ✹✳✷✻✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛✿ f (2f (x) − y) = 2x + y + 6, ∀x ∈ R ●✐↔✐✳ ❉➵ t❤➜② f ✤ì♥ →♥❤✳ ❈❤♦ y = −2x✱ t❛ ✤÷đ❝✿ f (2f (x) + 2x) = 6, ∀x ∈ R ❙✉② r❛✿ f (2f (x) + 2x) = f (2f (y) + 2y), ∀x, y ∈ R ❤❛② 2f (x) + 2x = 2c, ∀x ∈ R, c = const ❉♦ ✤â✿ f (x) = −x + c✳ ❚❤❛② ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ❜❛♥ ✤➛✉✱ t❛ ✤÷đ❝ c = 2✳ ❱➟②✱ f (x) = −x + 2, x ∈ R✳ ❱➼ ❞ö ✹✳✷✼✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛✿ f (f (x) + 2y − 1) = x − y + 7, ∀x ∈ R ✻✻ ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✻✼ ❱➼ ❞ö ✹✳✷✽✳ ❚➻♠ ❤➔♠ f : R → R t➠♥❣ ♥❣➦t t❤ä❛✿ f (f (x) + y) = f (x + y) + 1, ∀x, y ∈ R ❱➼ ❞ö ✹✳✷✾✳ ❚➻♠ ❤➔♠ f : R → R ❧✐➯♥ tö❝ ✈➔ t❤ä❛✿ f (x + f (y)) = f (x) + y, ∀x, y ∈ R ❱➼ ❞ö ✹✳✸✵✳ ❚➻♠ ❤➔♠ f : R → R t❤ä❛✿ f (xf (y) + x) = xy + f (x), ∀x, y ∈ R ✹✳✻ ❙û ❞ư♥❣ ❝→❝ t➟♣ trị ♠➟t ●✐↔ sû ❤➔♠ ❝➛♥ t➻♠ f (x) ①→❝ ✤à♥❤ tr➯♥ t➟♣ M ♥➔♦ ✤â✳ ▲ó❝ ♥➔②✱ ✭✶✮ ✲ ❈❤å♥ ♠ët t➟♣ ✤✐➸♠ B trị ♠➟t tr➯♥ M ✭ t❤ỉ♥❣ t❤÷í♥❣ B = Q ❤♦➦❝ m t➟♣ ❝→❝ ✤✐➸♠ B = n , m ∈ Z, n ∈ N ✱ B = Q ∩ [a, b]✱✳✳✳✮ ✭✷✮ ✲ ❚➼♥❤ ❣✐→ trà ❝õ❛ ❤➔♠ f (x) tr➯♥ t➟♣ ♥➔② ✭✸✮ ✲ ❙❛✉ ✤â sû ❞ö♥❣ t➼♥❤ ❝❤➜t ❝õ❛ ❤➔♠ f (x) ✭❝❤➥♥❣ ❤↕♥ ♥❤÷ t➼♥❤ ❧✐➯♥ tư❝✱ ✤ì♥ ✤✐➺✉✱ ❜à ❝❤➦♥✱✳✳✳✮ ✤➸ s✉② r❛ ❧í✐ ❣✐↔✐✳ P❤÷ì♥❣ ♣❤→♣ ♥➔② ✤÷đ❝ sû ❞ư♥❣ tr♦♥❣ ❧í✐ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❈❛✉❝❤② ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❏❡♥s❡♥✳ ❳➨t ♣❤÷ì♥❣ tr➻♥❤ ❤➔♠ ❞↕♥❣✿ f (x + y) = F (f (x), f (y), f (x − y), x, y) ❙û ❞ư♥❣ ♣❤÷ì♥❣ ♣❤→♣ ♥➔②✱ t❛ ❝â ❝→❝ ủ ỵ s x = y = t➻♠ ❣✐→ trà f (0) q✉❛ ♣❤÷ì♥❣ tr➻♥❤✿ f (0) = F (f (0), f (0), f (0), 0, 0) ✲ ❚❤❛② y = x ✈➔ x ❧➛♥ ❧÷đt ❜ð✐ 2x, 3x, , (n − 1)x ✤➸ t➼♥❤ f (kx), k = 2, n t❤æ♥❣ q✉❛ f (x)✿ f (2x) = F (f (x), f (x), f (0), x, x) = F2 (f (x), x) f (3x) = F (f (2x), f (x), f (x), 2x, x) = F3 (f (x), x) f (4x) = F (f (3x), f (x), f (2x), 3x, x) = F4 (f (x), x) f (nx) = F (f ((n − 1)x), f (x), f ((n − 2)), (n − 1)x, x) = Fn (f (x), x) ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✻✽ ✲ ❚❤❛② x = my/n✿ f (my) = Fn (f (my/n), my/n) ⇔ Fm (f (y), y) = Fn (f (my/n), my/n) ✲ ●✐↔✐ f (my/n)✿ f (my/n) = F(f (y), y, m/n) ✲ ❈❤♦ y = 1✿ f (q) = F(f (1), 1, q), ∀q ∈ Q+ ✲ ❙û ❞ư♥❣ t➼♥❤ ❝❤➜t ❤➔♠ ✭❧✐➯♥ tư❝✱ ✤ì♥ ✤✐➺✉✱✳✳✳✮✿ f (x) = F(f (1), 1, x), ∀x ∈ R+ ✲ ❑❤✐ x < 0✱ ❣✐→ trà f (x) ✤÷đ❝ t➼♥❤ t❤ỉ♥❣ q✉❛ f (−x)✿ ❈❤♦ y = −x ✈➔ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ t❤❡♦ f (x)✿ f (0) = F (f (x), f (−x), f (2x), x, −x) = F (f (x), F(f (1), 1, −x), F2 (f (x), x), x, −x) ❙✉② r❛✿ f (x) = G(x), x < 0✳  ❱➟②✱ +   F(f (1), 1, x), x ∈ R f (x) = f (0), x=0   G(x), x tỗ t {rn } Q+ s rn → x✳ ▲ó❝ ✤â✱ ✈➻ f ❧✐➯♥ tư❝ ♥➯♥✿ f (x) = f ( lim rn ) = lim f (rn ) = lim (a.rn2 ) = ax2 n→∞ n→∞ n→∞ ❱ỵ✐ x < 0✱ t❤❛② y = −x✿ f (x) = f (−x), ∀x ∈ R ❤❛② f ❧➔ ❤➔♠ ❝❤➤♥ tr➯♥ R✳ ❱➟②✱ f (x) = ax2 , x ∈ R✳ ❱➼ ❞ö ✹✳✸✷✳ ❚➻♠ ❤➔♠ f : R → R ❧✐➯♥ tö❝ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x + y) = f (x) + f (y) + f (x)f (y), ∀x, y ∈ R ●✐↔✐✳ ◆❤➟♥ ①➨t✿ f (x) = 2f (x/2) + [f (x/2)]2 = (1 + f (x/2))2 − ≥ −1, ∀x ∈ R ◆➳✉ f (1) = −1 t❤➻ f (x) = f (x − + 1) = f (x − 1) + f (1) + f (x − 1)f (1) = −1, ∀x ∈ R ❳➨t f (1) > −1✳ ❚✐➳♣ t❤❡♦✱ ❝❤♦ y ♥❤➟♥ ❧➛♥ ❧÷đt x, 2x, ✿ f (2x) = 2f (x) − f (x)2 = (f (x) + 1)2 − f (3x) = f (2x) + f (x) + f (2x)f (x) = (f (x) + 1)3 − f (nx) = (f (x) + 1)n − ❚❤❛② x ❜ð✐ my/n✿ f (my) = (f (my/n) + 1)n − ⇔ (f (y) + 1)m − = (f (my/n) + 1)n − ⇔ f (my/n) = (f (y) + 1)m/n − ❈❤♦ y = ✈➔ ✤➦t c = f (1) + > t❛ ✤÷đ❝✿ f (r) = cr − 1, ∀r ∈ Q+ ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P Pì ợ x > tỗ t {rn } Q+ s rn x✳ ▲ó❝ ✤â✱ ✈➻ f ❧✐➯♥ tư❝ ♥➯♥✿ f (x) = f ( lim rn ) = lim f (rn ) = lim (crn − 1) = cx − n→∞ n→∞ n→∞ ❉♦ f ❧✐➯♥ tö❝ ♥➯♥ f (0) = 0✳ ❱ỵ✐ x < 0✱ t❤❛② y = −x✿ f (0) = f (x) + f (−x) + f (x)f (−x) ⇒ f (x) = −f (−x) = cx − 1 + f (−x) ❱➟②✱ ♥❣❤✐➺♠ ❝õ❛ ❜➔✐ t♦→♥✿ f (x) ≡ −1, x ∈ R f (x) = cx − 1, x ∈ R ✈ỵ✐ c ❧➔ ❤➡♥❣ sè ❞÷ì♥❣✳ ❱➼ ❞ư ✹✳✸✸✳ ❚➻♠ ❤➔♠ f : R → R ❧✐➯♥ tö❝ s❛♦ ❝❤♦✿ f (x + y) = f (x) + f (y) + 2xy, ∀x, y ∈ R ❱➼ ❞ö ✹✳✸✹✳ ❚➻♠ ❤➔♠ f : R → R ❧✐➯♥ tö❝ s❛♦ ❝❤♦✿ f (x + 2y) = f (x) + 4f (y) + 4xy, ∀x, y ∈ R ✹✳✼ ❍➔♠ ❦❤↔ ✈✐ ◆➳✉ ❤➔♠ ❝➛♥ t➻♠ ❦❤↔ ✈✐ t❤➻ t❛ ❝â t❤➸ ❧➜② ✤↕♦ ❤➔♠ t❤❡♦ tø♥❣ ❜✐➳♥ ❦❤✐ ❜✐➳♥ ❦❤→❝ ❝è ✤à♥❤✳ ❱➼ ❞ö ✹✳✸✺✳ ❚➻♠ ❤➔♠ f : R → R ❦❤↔ ✈✐ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x + y) = f (x) + f (y) + xy, ∀x, y ∈ R ●✐↔✐✳ ▲➜② ✤↕♦ ❤➔♠ t❤❡♦ tø♥❣ ❜✐➳♥✿ f ′ (x + y) = f ′ (x) + y f ′ (x + y) = f ′ (y) + x ❚ø ✤➙② s✉② r❛✿ f ′ (x) + y = f ′ (y) + x, ∀x, y ∈ R ⇔ f ′ (x) − x = f ′ (y) − y, ∀x, y ∈ R x2 ❉♦ ✤â✿ f (x) − x = a, a = const ❤❛② f (x) = + ax + b✳ x2 ❚❤û ❧↕✐✱ b = 0✳ ❱➟②✱ f (x) = + ax, x ∈ R✳ ′ ❈❍×❒◆● ✹✳ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✼✶ ❱➼ ❞ö ✹✳✸✻✳ ❚➻♠ ❤➔♠ f : R → R ❦❤↔ ✈✐ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ xf (y) + yf (x) , ∀x, y ∈ R ❜✮ f (x + 2y) = f (x) + 2f (y) + 5, ∀x, y ∈ R ❝✮ f (x + y) + f (x − y) = 2f (x)f (y), ∀x, y ∈ R ❛✮ f (xy) = ❱➼ ❞ö ✹✳✸✼✳ ❚➻♠ ❤➔♠ f : R+ → R ❦❤↔ ✈✐ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥✿ f (x) + f (y) √ ❛✮ f ( xy) = , ∀x, y > ❜✮ f (xy ) = f (x) + 2f (y), ∀x, y > ✹✳✶✳ ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✹ ▷ ❳→❝ ✤à♥❤ ❝→❝ ❤➔♠ sè f (x) ❧✐➯♥ tö❝ tr➯♥ ❘ t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥✿ ❛✮ f (2x) = f (5x), ∀x ∈ R✳ ❜✮f (3x) = f (5x) + 7x, ∀x ∈ R✳ ❝✮ f (x)(2 − f (x)) = x(2 − x), ∀x ∈ R✳ ❞✮ f (f (x)) = 3f (x) − 2x, ∀x ∈ R✳ ❡✮ f (2x − y) = 2f (x) − f (y), ∀x, y ∈ R✳ ❢✮ f (3x + y) = 3f (x) + f (y), ∀x, y ∈ R✳ ❣✮ f (x + y) + f (x − y) = 2f (x) + 2f (y), ∀x, y ∈ R✳ ❤✮ f (x + y)f (x − y) = [f (x)f (y)]2 , ∀x, y ∈ R✳ ✹✳✷✳ ▷ ❈❤♦ ❤➔♠ sè f : R → R✱ ❜à ❝❤➦♥ tr♦♥❣ ❧➙♥ ❝➟♥ ✤✐➸♠ x = ✈➔ t❤♦↔ ✤✐➲✉ ❦✐➺♥✿ ❛✮ f (x) = f (3x) + x2 , ∀x ∈ R✳ ❜✮ f (2x) = 5f (3x) − x2 , ∀x ∈ R✳ ▷ ✹✳✸✳ ❈❤♦ ❤➔♠ sè f : R → R t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ f (x) ≤ x, ∀x ∈ R, f (x + y) ≤ f (x) + f (y), ∀x, y ∈ R ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ f (x) ≡ x✳ ✹✳✹✳ ●✐↔ sû ❤➔♠ f : R → R t❤ä❛ ✤✐➲✉ ❦✐➺♥ ỗ t số K > s ❝❤♦ |f (x) − f (y)| ≤ K(x − y)2 , ∀x, y ∈ R ❈❤ù♥❣ ♠✐♥❤ r➡♥❣ f ≡ const✳ ❈❍×❒◆● ✹✳ ▷ ❈⑩❈ P❍×❒◆● P❍⑩P ●■❷■ P❍×❒◆● ❚❘➐◆❍ ❍⑨▼ ✹✳✺✳ ❳→❝ ✤à♥❤ ❤➔♠ f : R → R ❧✐➯♥ tö❝ ✈➔ t❤ä❛ ✤✐➲✉ ❦✐➺♥ f (x − y)f (y − z)f (z − x) + = 0, ∀x, y, z ∈ R ▷ ✹✳✻✳ ❚➻♠ ❝→❝ ❤➔♠ f (x) ①→❝ ✤à♥❤ ✈➔ ❦❤↔ ✈✐ tr➯♥ R t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ f (x + y) = f (x) f (y), ∀x, y ∈ R ✼✷ ❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦ ❬✶❪ ◆❣✉②➵♥ ❱➠♥ ▼➟✉ ✭✷✵✵✵✮✱ P❤÷ì♥❣ tr➻♥❤ ❤➔♠✱ ◆❳❇ ●✐→♦ ❞ư❝✳ ❬✷❪ ❚r➛♥ ▼✐♥❤ ❍✐➲♥ ✭✷✵✶✶✮✱ ❈❤✉②➯♥ ✤➲✿ P❤÷ì♥❣ tr➻♥❤ ❤➔♠ ✲ ❑ÿ t❤✉➟t ❣✐↔✐ ✈➔ ♠ët sè ✈➜♥ q P r Pữợ ❏✳ ❆❝③❡❧✱ ❍❛♥s❥♦r❣ ❖s❡r ✭✷✵✵✻✮✱ ▲❡❝t✉r❡s ♦♥ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❚❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❈♦✉r✐❡r ❈♦r♣♦r❛t✐♦♥✳ ❬✹❪ ❈❤r✐st♦♣❤❡r ●✳ ❙♠❛❧❧ ✭✷✵✵✼✮✱ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s ❛♥❞ ❍♦✇ t♦ ❙♦❧✈❡ ❚❤❡♠✱ ❙♣r✐♥❣❡r✳ ❬✺❪ Pr❛s❛♥♥❛ ❑✳ ❙❛❤♦♦✱ P❛❧❛♥✐❛♣♣❛♥ ❑❛♥♥❛♣♣❛♥ ✭✷✵✶✶✮✱ ■♥tr♦❞✉❝t✐♦♥ t♦ ❋✉♥❝t✐♦♥❛❧ ❊q✉❛t✐♦♥s✱ ❈❤❛♣♠❛♥ ✫ ❍❛❧❧✴❈❘❈✳ ✼✸ ... ❤②♣❡r❜♦❧✐❝✿ ex − e−x ✲ ❍➔♠ f (x) = sinh x = ✿ f (3x) = 3f (x) + 4[f (x)]3 , ∀x ∈ R ex + e−x ✲ ❍➔♠ f (x) = cosh x = ✿ f (x + y) + f (x − y) = 2f (x) f (y), ∀x, y ∈ R sinh x ex − e−x ✲ ❍➔♠ f (x) = x... (x + y) = f (x) + f (y) , ∀x, y ∈ R + f (x) f (y) cosh x ex + e−x ✲ ❍➔♠ f (x) = cosh x = ✿ = x sinh x e − e−x f (x + y) = + f (x) f (y) , ∀x, y : x, y, x + y ̸= f (x) + f (y) ❇➔✐ t➟♣ ❝❤÷ì♥❣ ✶