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Q. 86

Expert-verified

Calculus

Found in: Page 122

Book edition 1st

Author(s) Peter Kohn, Laura Taalman

Pages 1155 pages

ISBN 9781429241861

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Short Answer

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x = c$ .
$f (x) = \{\begin{aligned} 2 - x, if x rational \\ x^{2}, if x irrational, \end{aligned} c = 1$

Ans: $f (x) = \{\begin{array}{cc} 2 - x, & if x rational \\ x^{2}, & if x irrational \end{array}$ is continuous at point $c = 1$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1. Given information.

given expression, $f (x) = \{\begin{aligned} 2 - x, if x rational \\ x^{2}, if x irrational, \end{aligned} c = 1$

Step 2. Finding the limits at x=1.

LHL $= lim_{x \to 1^{-}} f (x) = lim_{x \to 1^{-}} 2 - x = 2 - 1 = 1$

RHL $= lim_{x \to 1^{+}} f (x) = lim_{x \to 1^{+}} x^{2} = 1^{2} = 1$

since, LHL = RHL

Therefore, $f (x)$ is continuous at a point $c = 1$

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