This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Chapter 4 Limits of Functions and Continuity 4.0 Some Definitions Below is our working definition of function . Informal definition . A function is a “rule” that assigns each element in a given set X a unique value in a set Y . The set X is called the domain of the function f (notation: X = dom( f )) and Y is called the codomain of f . We can express all of this at once using the compact notation f : X → Y . If x ∈ X is assigned to y ∈ Y , we can write y = f ( x ), where f represents the “rule.” The range of f is the set { y ∈ Y : f ( x ) = y for some x ∈ X } . We often denote the range by f ( X ) or ran( f ). The domain is simply the set from which the function takes its “input” values. The codomain is a set that contains potential “outputs” of the function. The range contains all and only the “output” values of f . We we specify a function, we must declare its domain; however, if a function has an obvious “natural” domain, then we can simply define the function without specifying the domain. In this case, the domain will be the “natural” domain on which the function (rule) makes sense. Similarly, if the codomain is not specified, we will simply assume that it is the range. Example 4.1. Let f ( x ) = √ x . Using the above convention, the domain of f is dom( f ) = { x ∈ R : x ≥ } and the range of f is ran( f ) = { y ∈ R : y ≥ } . (We will later show this last statement rigorously using the intermediate value theorem .) Since the codomain is not specified, we will assume that it is equal to the range. 41 Example 4.2. Let f : R → R be defined by f ( x ) = x 2 . This time, we have explicitly and fully defined a function f whose domain is R and whose codomain is R . The range of f is { y ∈ R : y ≥ } . (We will later show this last statement rigorously using the intermediate value theorem .) Here is the formal definition of a function. Definition 4.0.1. Given nonempty sets X and Y , a function from X into Y is a subset f of X × Y with the additional property that if ( x,y 1 ) ∈ f and ( x,y 2 ) ∈ f , then y 1 = y 2 , for all x ∈ X , y 1 , y 2 ∈ Y . Here, X is called the domain of f (notation: dom( f ) = X ) and Y is called the codomain of f . We can display all of this information together using the notation f : X → Y . If ( x,y ) ∈ f , we will write y = f ( x ). The set { y ∈ Y : ( x,y ) ∈ f for some x ∈ X } is called the range of f , denoted as f ( X ) or ran( f ). Definition 4.0.2. The graph of a function f : X → Y is the set of ordered pairs { ( x,f ( x )) : x ∈ X } . Under the formal definition of a function, the graph of f is just the set f itself. We say that two functions f and g are equal if they have the same graph; that is, f = g as sets....
View
Full
Document
This note was uploaded on 10/21/2010 for the course MATH 147 taught by Professor Wolzcuk during the Fall '09 term at Waterloo.
 Fall '09
 Wolzcuk
 Continuity, Limits

Click to edit the document details