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Q. 90

Expert-verifiedFound in: Page 122

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 $x\in R$

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

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

Let $c$ be any real number.

$f$ is continuous at $x=c$

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

if, $\underset{x\to c}{lim}\u200af\left(x\right)=f\left(c\right)$

$LHS=\underset{x\to c}{lim}\u200af\left(x\right)=\underset{x\to c}{lim}\u200a{x}^{3}$

by putting $x=c$

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

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

So, Function is continuous at $x=c$

Thus, we can write that

$f\left(x\right)={x}^{3}continuousforallx\in R$

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