Explanation Given information: A function is differentiable at point c. Concept used: If a function f ( x ) is differentiable at point c, then the following limit exist. lim h → 0 + f ( c + h ) − f ( c ) h and lim h → 0 − f ( c + h ) − f ( c ) h lim h → 0 + f ( c + h ) − f ( c ) h ≠ lim h → 0 − f ( c + h ) − f ( c ) h Formula Used: A function f ( x ) will be continuous at a point a if the following condition satisfies. lim x → a − f ( x ) = lim x → a + f ( x ) = f ( a ) Calculation: Now for check the left continuity of a function f ( x ) at point c, let us consider the following limit. lim x → c − [ f ( x ) − f ( c ) ] = lim x → c − [ ( x − c ) f ( x ) − f ( c ) x − c ] = lim x → c − ( x − c ) lim x → c − f ( x ) − f ( c ) x − c = ( c − c ) lim x → c − f ( x ) − f ( c ) x − c [ lim x → c − f ( x ) − f ( c ) x − c e x i s t ] = 0 Now for check the left continuity of a function f ( x ) at point c, let us consider the following limit

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Concept explainers

Rate of Change

The relation between two quantities which displays how much greater one quantity is than another is called ratio.

Slope

The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:

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Question

Chapter 3.2, Problem 5PQ

To determine

To show that a function is differentiable at a point c, then it is necessary to continuous at that point.

To find a continuous function at point c, but not differentiable.

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Chapter 3.2, Problem 4PQ

Chapter 3.2, Problem 1E

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Chapter 3 Solutions

Calculus - Standalone book

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To show that a function is differentiable at a point c, then it is necessary to continuous at that point. To find a continuous function at point c, but not differentiable.

Solution Summary: The author explains that to show that a function is differentiable at point c, it is necessary to continuous at that point.