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(2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies the conclusion of the Intermediate Value Theorem.

(2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies the conclusion of the Intermediate Value Theorem.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5.3

Please include a formal proof. Thanks!

Theorem 3.5.2 (Intermediate Value Theorem). Let [a,b] C R be a closed bounded interval, and let f : [a,b] → R be a function. Suppose that f is continuous. Let r E R. If r is strictly between f(a) and f(b), then there is some c E (a,b) such that f(c) =r.
Transcribed Image Text:Theorem 3.5.2 (Intermediate Value Theorem). Let [a,b] C R be a closed bounded interval, and let f : [a,b] → R be a function. Suppose that f is continuous. Let r E R. If r is strictly between f(a) and f(b), then there is some c E (a,b) such that f(c) =r.
Exercise 3.5.1. (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies the conclusion of the Intermediate Value Theorem.
Transcribed Image Text:Exercise 3.5.1. (2) Find an example of a function f : [0,1] → R such that f is not continuous, but that f satisfies the conclusion of the Intermediate Value Theorem.
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