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Chapter 10: Q. 93 (page 777)

Write a delta鈥揺psilon proof that proves that fis continuous on its domain. In each case, you will need to assume that 未 is less than or equal to 1.

f(x)=x1/2

Short Answer

Expert verified

Ans: f(x)=x12is continuous on its domain (continuous for all x鈭�R)

Step by step solution

01

Step 1. Given information.

Given, f(x)=x1/2

02

Step 2. Domain: 

Since, xis defined fo all x鈭�R

03

Step 3. So, check for continuity.  

Let cbe any real number.

fis continuous atx=c

assume thatc is less than or equal to1

x=cif, role="math" localid="1648053742246" limx鈫�c鈥�f(x)=f(c)

LHS=limx鈫�c鈥�f(x)=limx鈫�c鈥�xPuttingx=c=c

RHS=f(c)=c


04

Step 4.  Since, LHS=RHS

So, Function is continuous at x=c

Thus, we can write that

f(x)=x12continuous for allx鈭�R.

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