204Lecture52006 - Economics 204 Lecture 5Friday August 4...

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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 real-valued 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 Bolzano-Weierstrass Theorem, { t n } contains a convergent subsequence { t n k } , with lim k →∞ t n k = t 0 [ a, b ] Since f is continuous, f ( t 0 ) = lim t t 0 f ( t ) = lim k →∞ f t n k = M so M is finite and f ( t 0 ) = M = sup { f ( t ) : t [ a, b ] } so f attains its maximum and is bounded above. The argument for the minimum is similar. 1
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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 hands-on proof already. Now, we can simplify it a bit. Let B = { t [ a, b ] : f ( t ) < d } a B , so B = . 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 B , or t n = c = b , in which case f ( t n ) > d , so again t n 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 . Since 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 = c = b , so c ( a, b ). Monotonic Functions: Definition 3 A function f is monotonically increasing if y > x f ( y ) f ( x ) 2
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Theorem 4 (6.27) Suppose f is monotonically increasing on ( a, b ) . Then the one-sided limits f ( t + ) = lim s t + f ( s ) f ( t ) = lim s t f ( s ) exist and are real numbers for all t ( a, b ) . Proof: This is analogous to the proof that a bounded monotone sequence converges.
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