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Q. 88

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

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

See the step by step solution

Step 1. Given information.

given, $f\left(x\right)={x}^{-1}$

Step 2. Find Domain.

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

At $x=0$,

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

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

localid="1648049491586" $LHS=\underset{x\to c}{lim} f\left(x\right)=\underset{x\to c}{lim} \frac{1}{x}\phantom{\rule{0ex}{0ex}}Puttingx=c\phantom{\rule{0ex}{0ex}}=\frac{1}{c}$

localid="1648049934543" $RHS=f\left(c\right)=\frac{1}{c}$

Step 4. LHS=RHS

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

Thus, we can write that

$fiscontinuousforallx\in R-\left\{0\right\}$