The Physics Philes, lesson 27: Math, Continuously
In which conditions are met, a new concept is defined, and the author begs to be corrected.
After several weeks of limits, it’s finally time to move on. But for right now, it’s time for a new concept: continuity.
A function is continuous if it’s, well, continuous. OK, that’s not a helpful definition. Basically, as long as the function has no holes, breaks, or jumps, then it’s continuous. Limits told us the height a function intends to reach. Continuity guarantees the function made it to that height. There is a more precise mathematical definition of continuity. Basically, if a function is continuous, it must satisfy three conditions:
- The limit, as x approaches c, of f(x) exists
- f(c) is defined
- The limit, as x approaches c, of f(x) equals f(c)
So a function is continuous if the function has an intended height, there is no hole at that intended height, and the functions value matches its intended value. Now that we know the rules, let’s see how those rules are applied with an example. Let’s figure out if the following function is continuous at x = 1.
I’m just going to tell you right off the bat that I’m not sure if I’m doing this right, or if I understand the rational correctly. So if I get anything wrong, please please please let me know. OK. Let’s do this.
According to this piecewise-defined function, g(1) = -2. Now let’s use the factoring method to find the limit of the top rule as x approaches 1.
Now we can just plug in the 1 and see what we get.
As you see, as x approaches 1, g(x) equals 5. This means that the function we’ve been evaluating is discontinuous. Do you see why? We know that g(1) = -2. But our calculations also determined that as x approaches 1, g(x) is equal to 5. These values are not equal, so the third condition for a continuous function is not met.
A function isn’t just continuous or discontinuous. There are different kinds of discontinuity: jump, point, and infinite.
First, let’s look at jump discontinuity. The example we’ve already done is an example of jump discontinuity. This type of discontinuity is typically caused when a piecewise-defined function doesn’t connect neatly. You get jump discontinuity when no general limit exists at the given x-value because the right- and left-handed limits exist but are not equal.
The second type of discontinuity is point discontinuity. This is pretty much what it sounds like. Point discontinuity occurs when there is a hole in the function at a particular point. A general limit exists, but the function value is not defined there. It’s pretty easy to test for this type of discontinuity. Just test for a limit. Let’s look at the following function:
There is one, final type of discontinuity to discuss today, and that is infinite (or essential) discontinuity. This type of discontinuity occurs when a function neither has a limit nor is the limit defined at at given x-value. You’ll usually find this with vertical asymptotes. Since a vertical asymptote increases or decreases infinitely, there can be no limit. And this type of function never actually touches the asymptote, the function is undefined there. Do you remember how we test for this type of thing? We know we have a vertical asymptote if we get a constant divided by zero when we plug in an x-value.
That’s all the math from me for today, but before I skedaddle I have a request. While I think the book I’m using to teach myself calculus has been valuable in imparting the basics, I think I need more practice. Much more practice. Are there any good trig and calculus workbooks out there? Or any good websites with practice problems and detailed explanations? I feel like my understanding is relatively shallow so far. I want to wade a little into the deep end. Please leave your recommendations and corrections in the comments. They are always welcome.
Featured image credit: sashafatcat