{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# n4 - Atsushi Inoue ECG 765 Mathematical Methods for...

This preview shows pages 1–2. Sign up to view the full content.

Atsushi Inoue ECG 765: Mathematical Methods for Economics Fall 2009 Lecture Notes 4 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 x 0 be a point in the domain X of the function f . We say that f is continuous at x 0 if, for a given > 0, there exists a δ ( ) > 0 such that if x X and d ( x, x 0 ) = k x - x 0 k < δ ( ), then d ( f ( x ) , f ( x 0 )) = | f ( x ) - f ( x 0 ) | < . If f is continuous at every point of X , we say that f is continuous on X . Examples. f ( x ) = a + bx. g ( x ) = ( 0 if x < z 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 x 0 is a limit point ( accumulation point ) if every neighborhood of x 0 contains at least one point of S distinct from x 0 . 1

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Example. S = ( -∞ , - 1) ∪ { 0 } ∪ (1 , ). Definition. Let f : X → < and x 0 be a limit point of X . We write f ( x ) y 0 as x x 0 , or lim x x 0 f ( x ) = y 0 , if there is a point y 0 ∈ < with the following property: For every
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 2

n4 - Atsushi Inoue ECG 765 Mathematical Methods for...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online