Q. 86 Use the delta-epsilon definition... [FREE SOLUTION]

Chapter 1: Q. 86 (page 122)

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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

given expression, $f (x) = \{\begin{aligned} 2 鈭� x, 鈥呪赌呪赌呪赌� if x rational \\ x^{2}, 鈥呪赌呪赌呪赌� if x irrational, \end{aligned} c = 1$

Step 2. Finding the limits at x=1.

LHL $= lim_{x 鈫� 1^{鈭�}} 鈥� f (x) = lim_{x 鈫� 1^{鈭�}} 鈥� 2 - x = 2 - 1 = 1$

RHL $= lim_{x 鈫� 1^{+}} 鈥� f (x) = lim_{x 鈫� 1^{+}} 鈥� x^{2} = 1^{2} = 1$

since, LHL = RHL

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

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91影视

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Finding the limits at x=1.

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91影视

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point x=c.f(x)=2鈭�x,鈥呪赌呪赌呪赌�ifxrationalx2,鈥呪赌呪赌呪赌�ifxirrational,c=1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Finding the limits at x=1.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Mechanics Maths

Applied Mathematics

Calculus

Logic and Functions

Pure Maths

Study anywhere. Anytime. Across all devices.

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