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Daisy, track extraordinaire, is hiking the Appalachian Trail in central Pennsylvania. It's her dream to hike the entire trail from Georgia to Maine. It's been raining ever since she crossed the Mason-Dixon line, but that's not enough to make her quit. She hikes on, until she reaches a stream. The stream is far deeper than usual, and is way too dangerous to ford. There's a discontinuity in the trail so to speak. She doesn't have time to wait for the water to recede, and she's come too far to turn back. Daisy can handle any discontinuities this trail throws her way, though. She looks around for a sturdy tree limb and fashions a 14-foot pole. She uses it to successfully leap across the stream, and continues on her way. Boom. Pole vaulted.
We can also talk about the continuity or discontinuity of functions. If a function contains breaks, that function isn't continuous.
In this chapter we'll use graphs and formulas to describe continuity. Understanding continuity will also give us some freebie information about functions, and who doesn't like free stuff?
Continuity at a Point
This website gives you a few more examples of different functions and a discussion of continuity at different points. You’ll see some of the craziest examples that mathematicians can cook up. No, that doesn't include ramen and peanut butter sandwiches.
Limit of a Function and Continuity
This page discusses continuity and limits with a clear set of definitions and a number of examples. Like an oak tree, this page will firmly plant these ideas in your noggin. What, your oak trees don't do that?
Math Open Reference
If you've ever thought about the age-old question of applying the human genome project to function splicing, wonder no more: here's a description of the different types of discontinuities in functions.
Math Is Fun: Interval Notation
If you often ask yourself, "Do I use a square bracket or do I use a parenthesis?" (we do it all the time), this link will come to your rescue. With an unparalleled explanation of interval notation, you'll never need to ask yourself this question again.
Calculus-Help: The Intermediate Value Theorem
When you first saw the intermediate value theorem, it probably resembled the Russian language more than it did a mathematical idea. This video gives a very clear description of the IVT. Ура!
Intermediate Value Theorem Examples
You may think the intermediate value theorem is about as useful as a pogo stick in a grocery store. That's just silly. How else would you reach the top shelf? This video will show you some practical uses for the theorem…in math, anyway.
More Intermediate Value Theorem Examples
This video gives you a few more-complicated examples on how to use the intermediate value theorem. These are the questions your teacher may use to trick you. Don’t succumb.
Dr. X: Extreme Value Theorem
Let Dr. X explain the extreme value theorem to you in plain and simple terms. No, he won’t be able to help you with your mutant ability to grow out your fingernails at an inexplicably fast rate.
Checking Continuity of a Functions at a Point
This is the type of question your teacher will ask you to solve on the continuity exam, and this video will show you how to solve the problem and bag those 10 extra puntos.
Calculus Applets: The Extreme Value Theorem
This online tool helps teach the extreme value theorem. Unlike a gym membership, however, you can only use this theorem when the interval is closed.