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Expert-verified Found in: Page 122 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # 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 $1$. $f\left(x\right)={x}^{3}$

Ans: ${x}^{3}$ is continuous in its domain $x\in R$

See the step by step solution

## Step 1. Given information.

given expression $f\left(x\right)={x}^{3}$

## Step 2. Domain:

Since, $x$ is defined fo all $x\in R$

## Step 3. So, check for continuity.

Let $c$ be any real number.

$f$ is continuous at $x=c$

assume that $c$ is less than or equal to $1$

if, $\underset{x\to c}{lim} f\left(x\right)=f\left(c\right)$

$LHS=\underset{x\to c}{lim} f\left(x\right)=\underset{x\to c}{lim} {x}^{3}$

by putting $x=c$

localid="1648044466347" $={c}^{3}$

$RHS=f\left(c\right)={c}^{3}$

## Step 4. Since, LHS=RHS

So, Function is continuous at $x=c$

Thus, we can write that

$f\left(x\right)={x}^{3}continuousforallx\in R$ ### Want to see more solutions like these? 