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

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

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

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Finding the limits at x=1.

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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−x, ifxrationalx2, 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

Statistics

Theoretical and Mathematical Physics

Mechanics Maths

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Probability and Statistics

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