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continuity

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May 27, 2017

 continuity

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poyo

May 27, 2017
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  1. f ͸఺ a ΛؚΉ͋Δ۠ؒͰఆٛ͞Ε͍ͯΔؔ਺ͱ͠·͢ɻ͜ͷͱ ͖ f ͕ a Ͱ࿈ଓͰ͋Δ͜ͱɺͭ·Γ lim

    x→a f(x) = f(a) ͕੒ΓཱͭͨΊͷඞཁे෼৚݅͸ lim n→∞ xn = a ͱͳΔશͯͷ਺ྻ {xn } ʹରͯ͠ lim n→∞ f(xn) = f(a) Ͱ͋Δ͜ͱΛূ໌͠·͢ɻ
  2. 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 ʹऩଋ͍ͯ͠Ε͹Α͍͚ͩͳͷͰɺ೚ҙͰ͢ɻ
  3. े෼ੑ ೉͍͠ͷͰରۮ lim x→a f(x) f(a) ͳΒ͹ lim n→∞ xn

    = a ͕ͩ lim n→∞ f(xn) f(a) ͳΔ {xn } ͕ଘࡏ͢Δ Λࣔ͠·͢ɻ
  4. े෼ੑ 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 2ແݶݸબͿͷ͸͜Θ͍Ͱ͕͢ɺେৎ෉ͩͱ৴͡·͢ɻ
  5. े෼ੑ ·ͣ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