7-1-3-12 - A Transition to Advanced Mathematics by Smith,...

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Unformatted text preview: A Transition to Advanced Mathematics by Smith, et.al (7 th ed.) Clarification/Hints for 1.3 Exercise 12 Note: solutions to such a problem can vary. Let R + = R > = { x R : x > } . (12a) As stated in book. From calculus, write the symbolic form for the definition of f is continuous at a . (12a) Clarified statement of (12a). Let f : R R be a function. Let a R . Write a symbolic form of the definition of f is continuous at a . (12a) A solution. Let f : R R be a function. Let a R . Then, by definition, f is continuous at a precisely when: for each > 0 there exists > 0 such that if | x- a | < then | f ( x )- f ( a ) | < . Thus the definition of f is continuous at a can symbolically be written as: ( R + ) ( R + ) ( x R ) [ | x- a | < | f ( x )- f ( a ) | < ] . By years of convention, it can also be written as: ( > 0) ( > 0) ( x R ) [ | x- a | < | f ( x )- f ( a...
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