1.10 Exploring Types of Discontinuities

Discontinuities

🎥Watch: AP Calculus AB/BC - Continuity, Part II

This is the first topic dealing with continuity in unit 1. Until this point, our main focus was limits and how to determine them. However, a large part in finding and determining limits is knowing whether or not the function is continuous at a certain point. In topics 1.9 - 1.13, we will discuss continuity and different types of discontinuities you will see on the AP Exam. 

There are four types of discontinuities you have to know: jump, point, essential, and removable. 

Resources:

Jump Discontinuity ↔

If the left and right-handed limits aren’t equal, then the double-handed limit does not exist (DNE). In this scenario, we would have a jump discontinuity. The graph of that would look something like this:

Math Warehouse

As you can see, there is a break in the physical continuity of the graph at x. If the limit from both sides is different, how can we find a specific value that the function approaches? We cannot. Thus, the limit DNE.

Point Discontinuity 🕳

A point discontinuity occurs when the function has a “hole” in it from a mission point because the function has a value at that point that is “off the curve.”

Wikipedia

Essential Discontinuity ♾️

An essential discontinuity occurs when the curve has a vertical asymptote. This is also called an infinite discontinuity. 

Math Warehouse

Removable Discontinuity 🚫

This last discontinuity is fairly common. A removable discontinuity occurs when you have a rational expression with a common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is “removable.”