chapter1 - Chapter 1 Functions Limits Continuity and...

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1 Chapter 1 Functions, Limits, Continuity and Differentiability 1 Functions of a single real variable (p.157 – p.158, p.173 – p.176, p.193 – p.197) 1.1 Sets A set is a collection of distinct objects called elements or members of that set. For example, { } 1,2,3,4,5 A = is a set and a list of all its elements is given. In general, we use the notation { x|x processes certain properties}to denote a set of objects that share some common properties. Also, if e is an element of a set A , we write A e (read as e belongs to A ). Given two sets A and B , we say that A is a subset of E (denoted by A E ) if all elements of A belong to E . Example 1 We use the symbol R to denote the set which contains exactly all the real numbers. Also, the following symbols are frequently used to describe the corresponding subsets of real numbers ( a , b are two distinct real numbers): ( ) [ ] ( ) , { | } [ , ) { | } , { | } [ , ) { | } , { | } a b x a x b a b x a x b a b x a x b a x x a a x x a = < < = < = ∞ = −∞ = < R R R R R (The other subsets like ( ) ( ) ( , ], , ,( , ], , a b a a −∞ −∞ ∞ are defined similarly). These sets are usually called intervals . In our discussion, most of the sets we consider are intervals. , 1.2 Functions A function f is a rule of correspondence that associates with each object x in one set A (called the domain of f ) a unique value ( ) f x from a second set B (called the codomain of f ). The set of values so obtained is called the range of the function. f domain codomain range It is customary to write f as: : f A B
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2 In this course, we will mainly study those functions whose domains and codomains are subsets of the set of real numbers (which is denoted by R ). Moreover, when the rule for a function is given by an equation of the form ( ) y f x = (for example: 1 2 + = x y ), x is often called the independent variable and y the dependent variable . 1.3 Operations on functions Consider the functions f and g with formulas 2 1 ) ( 2 = x x f , 9 ( ) 1. g x x x = + We can make a new function of f g + , where ( )( ) ( ) ( ) 2 9 1 1. 2 x f g x f x g x x x + = + = + + Clearly, x must be a number which belongs to both the domains of f and g . Similarly, we can define the functions , f g fg and / f g as follows: ( )( ) ( ) ( ) f g x f x g x = ( )( ) ( ) ( ) fg x f x g x = × ( )( ) ( ) ( ) / f x f g x g x = (defined only for those x with ( ) 0 x g ) 1.4 Composition of functions Let : , : f A B g B C be two functions. We define the composite of g with f by ( )( ) ( ) ( ) . g f x g f x = D Note that the domain of this function f g D is A (and its codomain is C ). In general f g D and g f D (if both are defined) are two different functions. The following example will illustrate this fact: Example 2 Let ( ) 2 : , f f x x = R R and ( ) : , 1 g g x x = + R R Then R R : g f D is ( ) ( ) ( ) ( ) ( ) 2 ( ) 1 1 . f g x f g x f x x = = + = + D However, R R : f g D is ( )( ) ( ) ( ) ( ) 2 2 1 g f x g f x g x x = = = + D , which is different from g f D . , 2 Elementary functions In this section we will introduce different types of functions that are frequently used.
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