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Chapter 1: Q. 92 (page 122)

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)=x鈭�1/2

Short Answer

Expert verified

Ans: f(x)=x-12is continuous on its domain (continuous for allx鈭�R-{0}.

Step by step solution

01

Step 1. Given information.

given,

f(x)=x鈭�1/2

02

Step 2.   Domain:

f(x)=x-12=1x=10=鈭�

Hence, is not defined atx=0.

03

Step 3. So, check for continuity at all points except 0.

Letc be any real number except 0.

assume that cis less than or equal to 1

fis continuous at x=c

if,limx鈫�c鈥�f(x)=f(c)

localid="1648053903983">LHS=limx鈫�c鈥�f(x)=limx鈫�c鈥�1xPuttingx=c=1c

localid="1648054533319" RHS=f(c)=1c

04

Step 4. Since, LHS = RHS

The function is continuous at x=c(Except0)

Thus, we can write that

fis continuous for allx鈭�R-{0}

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