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Q. 88

Found in: Page 122


Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861

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Short Answer

Write a delta–epsilon proof that proves that f is continuous on its domain. In each case, you will need to assume that δ is less than or equal to 1.


Ans: f(x)=x-1 is continuous on its domain (continuous for all role="math" localid="1648040276972" x R- {0})

See the step by step solution

Step by Step Solution

Step 1. Given information.

given, f(x)=x1

Step 2. Find Domain.

f(x)=x-1 =1x

At x=0,

f(0) =10 =

Hence, is not defined at localid="1648042677677" role="math" x=0

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

Let c be any real number except 0.

assume that c is less than or equal to 1

f is continuous at x=c

if, limxcf(x)=f(c)

localid="1648049491586" LHS = limxcf(x) =limxc1xPutting x=c =1c

localid="1648049934543" RHS =f(c) =1c

Step 4. LHS=RHS 

The function is continuous at x=c (Except 0)

Thus, we can write that

f is continuous for all x R- {0}

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