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Chapter 1: Q. 90 (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)=x3

Short Answer

Expert verified

Ans: x3is continuous in its domain x鈭�R

Step by step solution

01

Step 1. Given information.

given expressionf(x)=x3

02

Step 2. Domain: 

Since, xis defined fo all x鈭�R

03

Step 3. So, check for continuity. 

Let c be any real number.

fis continuous at x=c

assume that c is less than or equal to1

if, limx鈫�c鈥�f(x)=f(c)

LHS=limx鈫�c鈥�f(x)=limx鈫�c鈥�x3

by putting x=c

localid="1648044466347" =c3

RHS=f(c)=c3

04

Step 4. Since, LHS=RHS

So, Function is continuous at x=c

Thus, we can write that

f(x)=x3continuousforallx鈭�R

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