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# Ã¨Â§Â£Ã¦ÂžÂÃ¥Â­Â¦I Lecture3

Ã¨Â§Â£Ã¦ÂžÂÃ¥Â­Â¦I Lecture3

## (6) f(x) [a, b] f(x)

(6) f(x) [a, b] f(x) [a, b] ♣ sup f(x) x ∈ [a, b] . ∃ {x n } (x n ∈ [a, b]) f(x n ) → +∞ . Bolzano-Weirstrauss () ∃ {x np } ({x n } ) x np → α a ≤ x np ≤ b a ≤ α ≤ b ⇒ α ∈ [a, b] f f(x np ) → f(α) f(x np ) → +∞ f(x) sup f(x) = M(x ∈ [a, b]) sup m ∈ N() f(x m ) > M − 1 m x m . {x n } {x n } (a ≤ x n ≤ b) B-W ∃ {x np } x np → c a ≤ x np ≤ b a ≤ c ≤ b ⇒ c ∈ [a, b] ⇒ (f(c) ) f(x np ) > M − 1 n p f(x np ) ≤ M p → +∞ lim f(x n p ) = f(c) = M p→+∞ ⇒ f(x) x = c () Remark [a, b] (p.31 )

(7) 3.6 x M ∃ {x n } ∈ M (x n ≠ x) x n → x(n → ∞) 3.7 M∪ { M } = M M (closure) 3.8 M ⇔ M = M {x n } ∈ M x n → α ⇒ α ∈ M . 3.5 f(x) I ∀ ε ∃ δ(ε ) > 0 |x − x ′ | < δ(x, x ′ ∈ I) ⇒ |f(x) − f(x ′ )| < ε Remark f(x) I ∀ a ∈ I ∀ ε ∃ δ(ε a ) |x − x ′ | < δ(x, x ′ ∈ I) ⇒ |f(x) − f(x ′ )| < ε

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