Question

All the points of discontinuity of the function $f$ defined by
$f(x)=\{\begin{array}{ll}3& \text{if }0<x<1\\ 4& \text{if }1<x<3are\\ 5& \text{if }3\le x\le 10\end{array}$

• A. 1 , 3
• B. 3 , 10
• C. 1 , 3 , 10
• D. 0 , 1 , 3

Answer 1

Answer: (A)

$f(x)=\{\begin{array}{ll}3& \text{if }0<x<1\\ 4& \text{if }1<x<3\\ 5& \text{if }3\le x\le 10\end{array}$
For $0\le x\le 1,f(x)=3;1<x<3;f(x)=4$ and
$3\le x\le 10,f(x)=5$ are constant functions, so it is continuous in the above interval,
so we have to check the continuity at $x=1,3$
At $x=1$, LHL $=\underset{x\to 1^{-}}{\lim }f(x)=\underset{x\to 1^{-}}{\lim }(3)=3$
$RHI=\underset{x\to 1^{+}}{\lim }f(x)=\underset{x\to 1^{+}}{\lim }(4)=4$
$\therefore LHL\ne RHL$
Thus, $f(x)$ is discontinuous at $x=1$
At $x=3,LHL=\underset{x\to 3^{-}}{\lim }f(x)=\underset{x\to 3^{-}}{\lim }(4)=4$
$RHL=\underset{x\to 3^{+}}{\lim }f(x)=\underset{x\to 3^{+}}{\lim }(5)=5$
$\therefore LHL\ne RHL$
Thus, $f(x)$ is continuous everywhere except at $x=1,3$