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Q. 91

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 $1$.

$f\left(x\right)={x}^{-2}$

Ans: $f\left(x\right)={x}^{-2}$ is continuous on its domain (continuous for all $x\in R-\left\{0\right\}$)

given, $f\left(x\right)={x}^{-2}$

$f\left(x\right)={x}^{-2}=\frac{1}{{x}^{2}}$

At $x=0,$

$f\left(0\right)=\frac{1}{0}=\infty $

Hence, is not defined at $x=0$

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

localid="1648051618930" $LHS=\underset{x\to c}{lim}\u200af\left(x\right)=\underset{x\to c}{lim}\u200a\frac{1}{{x}^{2}}\phantom{\rule{0ex}{0ex}}Puttingx=c\phantom{\rule{0ex}{0ex}}=\frac{1}{{c}^{2}}$

localid="1648051960287" $RHS=f\left(c\right)=\frac{1}{{c}^{2}}$

The function is continuous at $x=c(Except0)$

Thus, we can write that

$f$ is continuous for all localid="1648052075809" $x\in R-\left\{0\right\}$

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