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 ε > 0 there exists δ > 0 such that if  x a  < δ then  f ( x ) f ( a )  < ε . Thus the definition of “ f is continuous at a ” can symbolically be written as: ( ∀ ε ∈ R + ) ( ∃ δ ∈ R + ) ( ∀ x ∈ R ) [  x a  < δ ⇒  f ( x ) f ( a )  < ε ] . By years of convention, it can also be written as: ( ∀ ε > 0) ( ∃ δ > 0) ( ∀ x ∈ R ) [  x a  < δ ⇒  f ( x ) f ( a...
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This note was uploaded on 12/13/2011 for the course MATH 300 taught by Professor Staff during the Fall '11 term at South Carolina.
 Fall '11
 Staff
 Calculus, Advanced Math

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