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Each function is discontinuous at $x=a$. If the discontinuity may be removed, redefine the function so as to make it continuous at $x=a$ and decide if the function as redefined is differentiable at $x=a.$$$f(x)=\left\{\begin{aligned}x^{2} & \text { if } x<2 \\3 & \text { if } x=2 \quad a=2 \\6-x & \text { if } x>2\end{aligned}\right.$$

$f(2)=4$ not diff.

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Campbell University

Oregon State 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.

03:09

Find all discontinuities o…

03:44

Find the values of $x$ (if…

one. This function has a discount in eight and one that's gonna be given from the Ellen X squared him because X is never is continuing and has only just continuous. Ah, X is equal. Teoh Zero, we'll get. And that's an infinite this continuity, not a room one. Let's see. Well, actually, we have Thanks. Tellin explain, going it as this exit churches, you know, of a to not seen zero time negative infinity. So we're gonna need to take some nerves. Well, um, I don't want me Nothing, because our function is I say it will be necessary in our function is the kind that all rights not to find them. But it's like the ruble. And then So we're trying to determine what the limit exists. It'll be removal. So I see we're taking the lead. Let's say from the rain now, this function were to find that. Okay, let's see. Sloppy tones, role and all. God, one over on, uh, one C X word Teoh. We're taking conservative the the X. So again, 1/2 1 over X. Well, that's just X over two. And the limit, as this is virtually zero, all right, is evil Teoh. Let's see, we were taking element. Don't forget taking on it That's equal to zero him and it will be the same thing from because we're squaring or anything. We're approaching zero. Well, science, so I function can be continuous. Ah, we define it as excel in X squared one X is not equal to zero and just zero one X is equal to zero.

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