Q. 86 - Calculus

Question

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

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

Step-by-Step Solution

Answer

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 $c=1$

Step 1. Given information.

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

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 $c=1$