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Chapter 1: Q. 28 (page 107)
Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.
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Sketch a labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem, and illustrate on your graph that the conclusion of the Intermediate Value Theorem does not necessarily hold.
Use what you know about one-sided limits to prove that a function is continuous at a point if and only if it is both left and right continuous at .
Write delta-epsilon proofs for each of the limit statements in Exercises .
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Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.
Write a delta–epsilon proof that proves that f is continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1.
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