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Discuss (i) the continuity and; (ii) the differentiability of the given function. (iii) If a discontinuity is removable, redefine the function so as to make it continuous there. (iv) Is it now differentiable?$$\text { (a) } f(x)=\left\{\begin{array}{ll}x & \text { if } x<1 \\x^{2} & \text { if } x \geq 1\end{array} \text { (b) } g(x)=\left\{\begin{array}{ll}x^{3} & \text { if } x<2 \\x^{2} & \text { if } x \geq 2\end{array}\right.\right.$$

(a) (i) continuous everywhere (ii) not diff. at $x=1$(b) (i) non-remov, discont at $x=2$ not diff. there either

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Campbell University

University of Nottingham

Boston College

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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