Suggested languages for you:

Americas

Europe

Q. 86

Expert-verifiedFound in: Page 122

Book edition
1st

Author(s)
Peter Kohn, Laura Taalman

Pages
1155 pages

ISBN
9781429241861

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x=c$.

$f\left(x\right)=\left\{\begin{array}{r}2-x,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\text{rational}\\ {x}^{2},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\text{irrational},\end{array}\phantom{\rule{1em}{0ex}}c=1\right.$

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 $c=1$

given expression, $f\left(x\right)=\left\{\begin{array}{r}2-x,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\text{rational}\\ {x}^{2},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{if}x\text{irrational},\end{array}\phantom{\rule{1em}{0ex}}c=1\right.$

LHL $=\underset{x\to {1}^{-}}{lim}\u200af\left(x\right)=\underset{x\to {1}^{-}}{lim}\u200a2-x=2-1=1$

RHL $=\underset{x\to {1}^{+}}{lim}\u200af\left(x\right)=\underset{x\to {1}^{+}}{lim}\u200a{x}^{2}={1}^{2}=1$

since, LHL = RHL

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

94% of StudySmarter users get better grades.

Sign up for free