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Unformatted text preview: Atsushi Inoue ECG 765: Mathematical Methods for Economics Lecture Notes 4 Fall 2009 Continuous Functions and Existence of Optimal Solutions We will present sufficient conditions for an optimization problem to have a solution. Outline A. Continuous Functions B. Existence of Optimal Solutions A. Continuous Functions Definition. Let x0 be a point in the domain X of the function f . We say that f is continuous at x0 if, for a given > 0, there exists a ( ) > 0 such that if x X and d(x, x0 ) = x  x0 < ( ), then d(f(x), f (x0 )) = f (x)  f (x0 ) < . If f is continuous at every point of X, we say that f is continuous on X. Examples. f (x) = a + bx. 0 if x < z g(x) = 1 if x z Definition. (a) If x is a point in S, then any set which contains an open set containing x is called a neighborhood of x. (b) A point x0 is a limit point (accumulation point) if every neighborhood of x0 contains at least one point of S distinct from x0 . 1 Example. S = (, 1) {0} (1, ). Definition. Let f : X as x x0 , or if there is a point y0 is a > 0 such that and x0 be a limit point of X. We write f (x) y0
xx0 lim f (x) = y0 , > 0 there with the following property: For every f (x)  y0  < for all points x X \ {x0 } for which x  x0 < . "f (x) can be made arbitrarily close to y0 by taking x sufficiently close to x0 ." Example. f (x) = ex . Theorem. Assume that x0 is a limit point of X. If f is continuous at x0 if and only if limxx0 f (x) = f (x0 ). B. Existence of Optimal Solutions Weierstrass Theorem. Suppose f : X is continuous and X is nonempty and compact (closed and bounded). Then there exist x , x X such that f (x ) f (x) f (x ) for all x X. Suggested Exercises. Read Sections 12.35 and 13.4 of Simon and Blume (1994) and solve the exercises in these sections. Exercise. Define f : by 0 if x is a rational number 1 otherwise f (x) = Is this function continuous? 2 ...
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