 In this video, we provide the solution to question number 15 for the practice final exam for math 1210. We're given a function by its graph, its illustration is right here, and we're supposed to determine all the values of x where f is not continuous. That is, we're looking for discontinuities. Now, the things that immediately grabbed me is if I look at the y-axis, I see that there is a vertical asymptote occurring at x equals 0. That although the function is defined and the right-handed limit is going to be y equals 1, the left-handed limit is going off towards positive infinity. That indicates we have a vertical asymptote. That is a type of discontinuities. There's another discontinuity happening here at x equals 4. You'll see that the left-handed limit wants to be y equals 3, but the right-handed limit wants to be y equals 0. And so this is an example of a jump discontinuity. Even though the function is defined, the limit is undefined because the left and right-handed limits disagree with each other. Now, some other things that might catch your attention that are not discontinuities are the following. If you look on the left of the graph, there is a horizontal asymptote right here. We get that there's a horizontal asymptote at y equals 1. It turns out if you take the limit as x approaches negative infinity, the limit is going to become 1. That's not a discontinuity. That's just describing its in-behavior. Another thing that might distract you is the point on the right here at x equals 7. That's just the end of the domain. The function just suddenly stops at x equals 7. There's no discontinuity here. The function is defined to be y equals negative 2 at this location. And if you take the limit from the left as you approach x equals 7, you would also get y equals negative 2. Because there's nothing to the right, there's no disagreement. The limit here as x approaches 7 from the left or the right would be negative 2 because there is no right approach, so the limit is defined. So there's no discontinuity at the end point there as well. The other thing, and this is usually the biggest red herring to many students, is this corner or cusp that's occurring at x equals 2. The concern here is that a cusp, this sharp corner indicates that the function is not differentiable at x equals 2. But there's nothing wrong with continuity. I can draw this picture without picking up my pen at x equals 2. It's okay if you pause your drawing to switch directions, but you don't have to pick up your pen like you do at x equals 4, x equals 0 and x equals 4. So there's nothing wrong with 2. It turns out the discontinuities are at 0 and 4. So the correct answer is going to be choice E.