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Decide whether the function whose graph is shown is continuous. If it is not continuous, identify the $x$ -values at which it is discontinuous and classify the discontinuity.Figure 11

Removable discontinuity at $x=1$ and $2,$ non-removable at $x=3$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Campbell University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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Decide whether the functio…

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in problem per denying we had to check whether the function is continuous or not is the figure 11 represents the graph of the function. So for ah, function, Toby, continuous, it stays that I can't. A curve is continuous if it has no holes, bags or them towards Yeah, now figure alone. You can see that from figuring no one. We can see that at X is equal to yeah and yeah, X is equal to one and to and to and access equal to toe. There are old's if a girl of his whole. It means that this function is not defined at this particular value off X. It means that if off one and f off to does not exist, we cannot find the value off F one and f of to some type of this continued is called removable disk and continue t When we went off Scarface hole, we can feel Della We can fill that hole to remove the discontinuity. So this type of continued is called removable, Mhm discontinuity, right. Mhm. By filling the hole, we can remove the discontinuity so that the function is continuous. Now again at again at X is equal to three. There is a break. There is a break and also ah hole. The graph shows that had accessible to three. At exit is a call to cheers, a break and the whole. If a graph has break, it means that left hand side, left hand side limit, it's not equal to right hand side limit. We know that we know that for a function Toby continuous, there are three conditions that the function there are three conditions that is fo see is defined where sees any point in the domain off the function and limit extends to see Air Force X must exist, and the third condition is, and the third condition is limit. X approaches to see f off X should be equal to people. See if any functions satisfy these three conditions. Then they're functioning skull continuous function. So if, uh, growth off a function has hold, it means that FFC is not defined. Similarly, if our graph off the function is if a graph of a function has a break or a jump, it means that left hand side limit is not equal to right hand side limit. We chose that limit does not exist. So so if limit does not exist, they're discontinued, is called non removable discontinuity known remote and, well, discontinuity. But if my figure half a graph has only hold, that is that can be removed. So we call it is removable discontinuity.

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