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Chapter 1: Q. 18 (page 120)

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.

Short Answer

Expert verified

A labeled graph of a function that fails to satisfy the hypothesis of the Intermediate Value Theorem is

Step by step solution

01

Step 1. Given Information.

The Intermediate Value Theorem states that if f is continuous on a closed interval [a, b], then for any K strictly between f(a) and f(b), there exists at least one $c \in (a, b)$ such that $f (c) = K .$

02

Step 2. Sketching a graph

The graph is

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Most popular questions from this chapter

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$

$\underset{x \to - 1}{l i m} \frac{x^{2} - 2 x - 3}{x + 1} = - 4, ε = 0.1$

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to 0} (3 x^{2} + 1) = 1$ .

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.

$\lim_{x \to 0^{-}} f (x) = - 1, \lim_{x \to 0^{+}} f (x) = 1, f (0) = 0 .$

Write each of the inequalities in interval notation:

$0 < | x - 2 | < 0.1$

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .

$\lim_{x \to 3} \frac{1}{x} = \frac{1}{3}; you may assume δ \leq 1$

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