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continuity
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poyo
May 27, 2017
Education
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continuity
poyo
May 27, 2017
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Transcript
ྻͷۃݶͱؔͷۃݶ May 27, 2017
f a ΛؚΉ͋Δ۠ؒͰఆٛ͞Ε͍ͯΔؔͱ͠·͢ɻ͜ͷͱ ͖ f ͕ a Ͱ࿈ଓͰ͋Δ͜ͱɺͭ·Γ lim
x→a f(x) = f(a) ͕ΓཱͭͨΊͷඞཁे݅ lim n→∞ xn = a ͱͳΔશͯͷྻ {xn } ʹରͯ͠ lim n→∞ f(xn) = f(a) Ͱ͋Δ͜ͱΛূ໌͠·͢ɻ
ඞཁੑ lim x→a f(x) = f(a) ͱ͠·͢ɻͭ·Γɺ ∀ε, ∃δ (|x
− a| < δ ⇒ | f(x) − f(a)| < ε) ͕Γཱͭͱ͠·͢ɻ·ͨɺxn → a ΑΓ ∀ε1 , ∃N (N < n ⇒ |xn − a| < ε1) ͕Γཱ͍ͬͯ·͢ɻε1 ҙͳͷͰɺಛʹ্ͷࣜͷ δ ʹରͯ͠ ε1 = δ ͱ͢Δͱ ∀ε, ∃δ, ∃N (N < n ⇒ |xn − a| < δ ⇒ | f(xn) − f(a)| < ε) ͕ΓཱͭͷͰඞཁੑ͕ݴ͑·ͨ͠ɻ1 1{xn } a ʹऩଋ͍ͯ͠ΕΑ͍͚ͩͳͷͰɺҙͰ͢ɻ
ेੑ ͍͠ͷͰରۮ lim x→a f(x) f(a) ͳΒ lim n→∞ xn
= a ͕ͩ lim n→∞ f(xn) f(a) ͳΔ {xn } ͕ଘࡏ͢Δ Λࣔ͠·͢ɻ
ेੑ f a Ͱ࿈ଓͰͳ͍ͷͰ ∃ε1 , ∀δ, ∃x (|x
− a| < δ, |f(x) − f(a)| ≥ ε1) ͕Γཱ͍ͬͯ·͢ɻδ ҙͳͷͰɺͨͱ͑ δ = 1, 1/2, . . . , 1/n, . . . ʹରͯͦ͠ΕͧΕଘࡏ͢Δ x Λॱʹ x1 , x2 , . . . , xn , . . . ͱͯ͠ {xn } ΛఆΊ·͢ɻ2 ͜ͷͱ͖ɺ lim n→∞ xn = a lim n→∞ f(xn) f(a) ͕Γཱͪ·͢ɻҎԼʹ͜ΕΛࣔ͠·͢ɻ 2ແݶݸબͿͷ͜Θ͍Ͱ͕͢ɺେৎͩͱ৴͡·͢ɻ
ेੑ ·ͣɺN Λ N > 1/ε ʹͱΕ ∀ε, ∃N (N
< n ⇒ |xn − a| < ε) ͕Γཱͪ·͢ɻ࣮ࡍɺ{xn } ͷఆٛʹΑͬͯશͯͷ n ʹରͯ͠ |xn − a| < 1/n ͳͷͰ |xn − a| < 1 n < 1 N < ε Ͱ͢ɻ࣍ʹɺε = ε1 ͱ͢Ε ∃ε, ∀N (N < n, |f(xn) − f(a)| ≥ ε) ͕Γཱͪ·͢ɻ͜Ε {xn } ͷఆ͔ٛΒ໌Β͔Ͱ͢ɻ