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(a) From the graph of $ f $, state the numbers at which $ f $ is discontinuous and explain why.(b) For each of the numbers stated in part (a), determine whether $ f $ is continuous from the right, or from the left, nor neither.

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03:08

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Campbell University

Baylor University

University of Nottingham

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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(a) From the graph of $ f …

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So in this problem we're given this graph and asked to determine which numbers this graph is discontinuous and then asked him party to determine if they're continuous from the right from the left or neither. Okay, so we can see first of all that x equals -2 right here. All right, mm. The limit As X approaches -2 from the left is not equal to the limit as X approaches -2 from the right, but as we come in from the left here and from the right here we don't get to the same number. So we have a discontinuity right there. So we're discontinuous there, we are continuous from the left. We get closer and closer and closer we do have values of the function from the left And we are continuous from the right as well all the way up to -2. So we are continuous um both left and right. Just the limits are not the same. Okay, now where is the next dis continuity that we see? Well at X equals zero, it's still continuous as the graph just makes a turn right there but it is continuous all the way through and we have another one right here At x equals two. So at x equals two. The limit that exit and then exit will purchase two from the left again is not equal to the limit as X approaches to from the right. So we're not continuous there we are continuous from both the left and the right as we can see on the graph. This is a continuous function as we come into the from the left side and as we come in from the right side as well. Okay, then we have one more dis continuity here at X equals four. Right X equals four Because we can see that the limit as X approaches four from the left is minus infinity, which is not equal to the limit as X approaches for from the right. Okay. And we can still see that we are continuous from both the left and the right side as well from the right here, and as we approach in from the left, this continuous function discontinues onward. Mm

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