Chapter 3 Proofs

# Chapter 3 Proofs - Pg 43 Example 3.1 For each number x...

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Pg 43, Example 3.1 – For each number x, define . Then the function is continuous. To verify this, we select a point x 0 in R, and we will show that the function is continuous at x 0 . Let be a sequence that converges to x 0 . By the sum and product properties of convergent sequences, Thus, is continuous at x 0 . Pg 43, Example 3.2 – Define for . Then the function is continuous. To verify this, we select a nonnegative number and let be a sequence of nonnegative numbers that converges to . But then the sequence converges to ; that is, Thus, is continuous at x 0 . Pg 44, Example 3.3 – Define the function by Prove that there is no point in R (domain) in which the function f is continuous. Pf: Let x 0 be in R (domain). For each natural number n, by Thm 1.11, we may choose a rational number, which we will label , in the interval and an irrational, which we will label , in the interval . But for each natural number n, and , so Since both of the sequences and converges to x 0 , it is not possible for

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Chapter 3 Proofs - Pg 43 Example 3.1 For each number x...

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