Chapter3 - 3 Limits of functions Continuity After introducing the basic notions on functions limits and continuity we will go on to Bolzanos theorem and

Chapter3 - 3 Limits of functions Continuity After...

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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
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