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Intro to Analysis in-class_Part_27

# Intro to Analysis in-class_Part_27 - Chapter 4 Continuous...

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Unformatted text preview: Chapter 4 Continuous functions 4.1 Concepts of continuity 4.1.1 Definitions Definition 4.1.1. 1 A function from a set D to a set R is a subset f ⊆ D × R for which each element d ∈ D appears in only one element ( d, · ) ∈ f . Write f : D → R . If ( x,y ) ∈ f , then we usually write f ( x ) = y . D is the domain of the function; the subset of R of elements x for which the function is defined. R is the range ; a subset of R which contains all the points f ( x ). Generally, assume R = R . Definition 4.1.2. A function is a rule of assignment x 7→ f ( x ), where for each x in the domain, f ( x ) is a unique and well-defined element of the range. f ( x ) = y means “ f maps x ∈ D to f ( x ) ∈ R ”. Definition 4.1.3. The image of f is the subset Im f := { y ∈ R . . . ∃ x ∈ D,f ( x ) = y } ⊆ R. The function f is surjective or onto iff Im f = R , that is, ∀ y ∈ R, ∃ x ∈ D,f ( x ) = y....
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Intro to Analysis in-class_Part_27 - Chapter 4 Continuous...

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