Chapter 1: Q. 85 (page 122)

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

Short Answer

Expert verified

Ans: $f (x) = \{\begin{array}{ll} 2 - x, if x rational \\ x^{2}, if x irrational, \end{array} c = 2$ is not continuous at $c = 2$

Step by step solution

Step 1. Given information.

given,

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

Step 2. Finding the limits at point c=2

LHL $= lim_{x \to 2^{-}} f (x) = lim_{x \to 2^{-}} 2 - x = 2 - 2 = 0$

RHL $=$ localid="1648025007516" $lim_{x \to 2^{+}} f (x) = lim_{x \to 2^{+}} x^{2} = 2^{2} = 4$ $= lim_{x \to 1^{-}} f (x) = lim_{x \to 1^{-}} 2 - x = 2 - 1 = 1$

Since LHL $\neq$ RHL

Therefore, $f (x) = \{\begin{array}{ll} 2 - x, if x rational \\ x^{2}, if x irrational, \end{array} c = 2$ is not continuous.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

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

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Finding the limits at point c=2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Geometry

Probability and Statistics

Applied Mathematics

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)=2−x, ifxrationalx2, ifxirrational,c=2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Finding the limits at point c=2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Decision Maths

Pure Maths

Geometry

Probability and Statistics

Applied Mathematics

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