## 2.5 Continuity

### Summary

The concept of continuity is formally defined in terms of limit. But then, in a kind of "turning of the tables," it is pointed out that continuity is often used in evaluating limits. Finally, the Intermediate Value Theorem and some of its consequences are discussed.

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.

### 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.

#### 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?

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