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Unformatted text preview: 3 Limits of functions, Continuity After introducing the basic notions on functions, limits and continuity, we will go on to Bolzano’s theorem, and the Intermediate Value Theorem (IVT) which follows from it, as well as the Extremal Value Theorem (EVT). 3.1 Functions Def A function f is a set of ordered pairs ( x,y ), with x,y ∈ R , such that no two (ordered pairs) have the same first member. By definition, the second member y is determined by x , one may unam biguously write y as f ( x ), where f denotes the assignment x 7→ y . The set of all x (for which f is defined) is called the domain of f , and the set of the corresponding y is called the image (or range ) of f . Notation : f : X → Y , where X is the domain, and Y contains the range. One can plot the ordered pairs { ( x,y = f ( x )) } (defining a function f ) in the Cartesian plane R 2 = { ( x,y )  x,y ∈ R } , and the resulting figure is called the graph of f . It will be useful to become aware of the graphs of a number of standard functions, such as the ones below. Examples : (i) The identity function : f ( x ) = x ; (ii) Constant function : f ( x ) = c , for all x ∈ R , with c a fixed real number; (iii) Linear function : f = ax + b , for constants a,b ; (iv) Polynomial function of degree n ≥ 0: f ( x ) = ∑ n j =0 a j x j , with a 1 ,...,a n ∈ R , a n ̸ = 0; (v) Upper semicircle function : f ( x ) = √ r 2 − x 2 , X = { x ∈ R  − r ≤ x ≤ r } , r : radius > 0; (vi) The integral part function : f ( x ) = [ x ], the largest integer not greater than x , X = R , Image( f ) = Z . 1 We will call a function f onetoone , or injective , iff for every y in the range, there is a unique x such that f ( x ) = y . The function f : X → Y is said to be onto , or surjective , iff Y is the image of f , i.e., for every y in Y , there is an x ∈ X such that y = f ( x ). Note that the linear function example above is injective on X = R iff a ̸ = 0, in which case it is onto all of R . Sometimes f is not an injective function on its natural domain X , but becomes one when restricted to a (large enough) subset X 1 of X . (It is always injective if restricted to one point, but this is not interesting!) For example, the upper semicircle function is injective on { x  ≤ x ≤ r } ; the graph of this restriction is a quarter circle of radius r (in the first quadrant of R 2 ). 3.2 Open, closed and compact subsets of R By an interval , we will mean a subset I of R such that if a,b are in I , then any number x between a and b is in I . Examples are, for a < b ∈ R , the open interval ( a,b ) = { x ∈ R  a < x < b } , the closed interval [ a,b ] = { x ∈ R  a ≤ x ≤ b } , the halfopen intervals [ a,b ) = { x ∈ R  a ≤ x < b } , ( a,b ] = { x ∈ R  a < x ≤ b } , the infinite open intervals ( a, ∞ ) = { x ∈ R  a < x < ∞} , ( −∞ ,b ) = { x ∈ R  − ∞ < x < b } , and so on....
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 Winter '08
 Borodin,A
 Topology, Continuity, Intermediate Value Theorem, Limits, Continuous function, Metric space, Inverse function, Compact space

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