Chapter 1: Q. 42 (page 120)

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.
$f (x) = \{\begin{matrix} \sqrt{- x}, i f x < 0 \\ 2, i f x = 0 \\ \sqrt{x}, i f x > 0 \end{matrix} .$

Short Answer

Expert verified

$T h e g i v e n f u n c t i o n i s n o t c o n t i n u o u s a t x = 0, i t h a s p o i n t d i s c o n t i n u i t y a n d i t i s n e i t h e r l e f t n o r r i g h t c o n t i n u o u s .$

Step by step solution

Step 1. Given Information.

Given the function:

$f (x) = \{\begin{matrix} \sqrt{- x}, i f x < 0 \\ 2, i f x = 0 \\ \sqrt{x}, i f x > 0 \end{matrix} a n d i t h a s i t' s b r e a k p o i n t a t x = 0 .$

Step 2. Finding the limits at the break point.

$A t x = 0, L H L = \lim_{x \to 0^{-}} f (x) = \lim_{x \to 0^{-}} \sqrt{- x} = \sqrt{- 0} = 0 . R H L = \lim_{x \to 0^{+}} f (x) = \lim_{x \to 0^{+}} \sqrt{x} = \sqrt{0} = 0 . f (0) = 2 . N o w, L H L = R H L \neq f (0) .$

Step 3. Finding the type of discontinuity.

Since the function is discontinuous at $x = 0$ from Step 2.

Now we know LHL = RHL, none is equal to f(0) and both exist then this type of discontinuity is point discontinuity.

And also, $L H L \neq f (0) a n d a l s o R H L \neq f (0)$ So, it is neither left continuous nor right 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.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break point.

Step 3. Finding the type of discontinuity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Mechanics Maths

Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.f(x)=-x,ifx<02,ifx=0x,ifx>0.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Finding the limits at the break point.

Step 3. Finding the type of discontinuity.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Probability and Statistics

Mechanics Maths

Statistics

Calculus

Decision Maths

Study anywhere. Anytime. Across all devices.