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Q. 86

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Found in: Page 122

### Calculus

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

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# Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x=c$.$f\left(x\right)=\left\{\begin{array}{r}2-x, \text{if}x\text{rational}\\ {x}^{2}, \text{if}x\text{irrational},\end{array}\phantom{\rule{1em}{0ex}}c=1\right\$

Ans: $f\left(x\right)=\left\{\begin{array}{cc}2-x,& \text{if}x\text{rational}\\ {x}^{2},& \text{if}x\text{irrational}\end{array}\right\$ is continuous at point $c=1$

See the step by step solution

## Step 1. Given information.

given expression, $f\left(x\right)=\left\{\begin{array}{r}2-x, \text{if}x\text{rational}\\ {x}^{2}, \text{if}x\text{irrational},\end{array}\phantom{\rule{1em}{0ex}}c=1\right\$

## Step 2. Finding the limits at x=1.

LHL $=\underset{x\to {1}^{-}}{lim} f\left(x\right)=\underset{x\to {1}^{-}}{lim} 2-x=2-1=1$

RHL $=\underset{x\to {1}^{+}}{lim} f\left(x\right)=\underset{x\to {1}^{+}}{lim} {x}^{2}={1}^{2}=1$

since, LHL = RHL

Therefore, $f\left(x\right)$ is continuous at a point $c=1$

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