2.5 Continuity

Summary

By the end of your studying, you should know:

• The definition of continuity at a point.
• How to apply the definition to a given function.
• The definition of a point of discontinuity.
• The definition of a continuous function.
• The meaning of a removable discontinuity, and the continuous extension of a function.
• A statement of the Intermediate Value Theorem (IVT).
• How to use the IVT.

On-screen applet instructions: h → 0, i.e. to let x → 1. Note the different effects if you approach from the right vs. the left.

Examples What is What is Is |x| continuous at 0? Consider the function the biggest integer smaller than or equal to x. What can you say about the continuity of f(x)? A hiker sets out at dawn to walk over a mountain and down the other side to a lake, a distance of x kilometers, and he reaches his destination exactly 12 hours later. He camps, then leaves at the same time the next morning and travels the same trail back, arriving at his starting place 12 hours later. Use the Intermediate Value Theorem to show that at some specific time, he was at the same place on his hike on both days. Applets Continuity of Functions Videos See short videos of worked problems for this section. Quiz Take a quiz. Exercises See Exercises for 2.5 Continuity (PDF). Work online to solve the exercises for this section, or for any other section of the textbook.

Resources on the Web Information on Newton Biographical data from St. Andrew's University's Web site Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics" Information on Leibniz Biographical data from St. Andrew's University's Web site Excerpt from W.W. Rouse Ball's "A Short Account of the History of Mathematics" Calculus Applications Project Intermath Continuity Kouba

Interesting Application Question: Which theorem allows us to conclude that if the balloon has a maximum height of 50 meters and a minimum height of 20 meters as it passes through the valley, then it must have gone through a point 37 meters above the valley floor? Why?

2.4 Limits at Infinity

Table of Contents

2.6 Tangent Lines and Their Slopes

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel