In Exercises 1–4, say whether the function graphed is continuo...

Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?

Accepted Answer

Welcome back, everyone. In this problem, consider the function geoffX on the interval open bracket negative to 6 closed bracket. Is GeoX continuous on this interval? And if not, where does it fail to be continuous? Here we have our graph, and for our answer choices, A says it's continuous, B says no, it's not continuous at X equals 6, C says no at X equals -2, and D, no at X equals -2 and 6. Now what does it mean for a function to be continuous? Well, on a graph, a function is continuous if there are no breaks, jumps, or holes in that interval. So for our interval here, are there any parts where there are breaks, jumps, or holes? Well, right away, there might be 3 parts on our graph that stand out to us, where X is -2, 4, and 6. So let's examine each of these. First, let's start with when X is -2. So we're going to G of -2. What do we notice? Well, here, notice that the function has an open circle at this y value. That means G of negative 2 is not defined at that point. This represents a jump discontinuity, OK? So then we can say that G-2 is not defined. In other words, G of X is not defined where X equals -2. Next, if we go to where X equals 4, at this point, there is a sharp peak, right? But the function does not break or have a hole. Notice that it goes this way and then it comes down that way, OK? So, because it doesn't break or have a hole, and the left hand limit, OK, and the right hand limit, both equal the same, OK. They're both 4 at this point, Y equals 4. Then that means they are, since they exist and are equal to the function value, then our function is continuous at X equals 4. Let's go to our final spot where X equals 6. Well, notice here at this point, the function has an open circle at this y value, meaning G6 is not defined at this point, OK? A solid dot here represents a different y value. That means G6 is assigned a different value. Thus, this represents a jump discontinuity. Therefore, the function is also not defined at G of 6, OK. Which means that the function fails to be continuous at X equals -2, and X equals 6 due to a hole and a jump discontinuity. Therefore, D is the correct answer. It is not continuous because of the jumps at X equals -2 and 6. Thanks a lot for watching everyone. I hope this video helped.

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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Key Concepts

Continuity of Functions

Types of Discontinuities

Evaluating Limits

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