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Unformatted text preview: Economics 204 Lecture 5–Friday, August 4, 2006 Revised 8/7/06, revisions marked by ** Section 2.6 (Continued) Properties of Real Functions Theorem 1 (6.23, Extreme Value Theorem) Let f be a continuous realvalued function on [ a, b ] . Then f assumes its minimum and maximum on [ a, b ] . In particular, f is bounded above and below. Proof: Let M = sup { f ( t ) : t ∈ [ a, b ] } If M is finite, for each n , we may choose t n such that M ≥ f ( t n ) ≥ M − 1 n (if we couldn’t make such a choice, then M − 1 n would be an upper bound and M would not be the supremum). If M is in finite, choose t n such that f ( t n ) ≥ n . By the BolzanoWeierstrass Theorem, { t n } contains a convergent subsequence { t n k } , with lim k →∞ t n k = t ∈ [ a, b ] Since f is continuous, f ( t ) = lim t → t f ( t ) = lim k →∞ f t n k = M so M is finite and f ( t ) = M = sup { f ( t ) : t ∈ [ a, b ] } so f attains its maximum and is bounded above. The argument for the minimum is similar. 1 Theorem 2 (6.24, Intermediate Value Theorem) Suppose f : [ a, b ] → R is continuous, and f ( a ) < d < f ( b ) . Then there exists c ∈ ( a, b ) such that f ( c ) = d . Proof: We did a handson proof already. Now, we can simplify it a bit. Let B = { t ∈ [ a, b ] : f ( t ) < d } a ∈ B , so B 6 = ∅ . By the Supremum Property, sup B exists and is real so let c = sup B . Since a ∈ B , c ≥ a . B ⊆ [ a, b ], so c ≤ b . Therefore, c ∈ [ a, b ]. We claim that f ( c ) = d . Let t n = min ⎧ ⎪ ⎨ ⎪ ⎩ c + 1 n , b ⎫ ⎪ ⎬ ⎪ ⎭ Either t n > c , in which case t n 6∈ B , or t n = c = b , in which case f ( t n ) > d , so again t n 6∈ B ; in either case, f ( t n ) ≥ d . Since f is continuous at c , f ( c ) = lim n →∞ f ( t n ) ≥ d (Theorem 3.5 in de la Fuente). Since c = sup B , we may find s n ∈ B such that c ≥ s n ≥ c − 1 n Since s n ∈ B , f ( s n ) < d . Sinc e f is continuous at c , f ( c ) = lim n →∞ f ( s n ) ≤ d (Theorem 3.5 in de la Fuente). Since d ≤ f ( c ) ≤ d , f ( c ) = d . Since f ( a ) < d and f ( b ) > d , a 6 = c 6 = b , so c ∈ ( a, b ). Monotonic Functions: Definition 3 A function f is monotonically increasing if y > x ⇒ f ( y ) ≥ f ( x ) 2 Theorem 4 (6.27) Suppose f is monotonically increasing on ( a, b ) . Then the onesided limits f ( t + ) = lim s → t + f ( s ) f ( t − ) = lim s → t − f ( s ) exist and are real numbers for all...
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 Summer '08
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 Economics

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