Question

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

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

Answer 1

Answer: (A)

$f(x) = \begin{cases} 3 & \quad \text{if }\,\,\,\,0< x< 1 \\ 4 & \quad \text{if } \,\,\,\,1< x< 3 \\ 5 & \quad \text{if } \,\,\,\,3 \leq x \leq 10 \end{cases} $
For $0 \leq x \leq 1, f(x)=3 ; 1<\,x<\,3 ; f(x)=4$ and
$3 \leq x \leq 10, f ( x )=5$ 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 $=\displaystyle\lim _{x \rightarrow 1^{-}} f(x)=\displaystyle\lim _{x \rightarrow 1^{-}}(3)=3$
$RHI =\displaystyle\lim _{ x \rightarrow 1^{+}} f ( x )=\displaystyle\lim _{ x \rightarrow 1^{+}}(4)=4$
$\therefore LHL \neq RHL$
Thus, $f ( x )$ is discontinuous at $x =1$
At $x =3, LHL =\displaystyle\lim _{ x \rightarrow 3^{-}} f ( x )=\displaystyle\lim _{ x \rightarrow 3^{-}}(4)=4$
$RHL =\displaystyle\lim _{ x \rightarrow 3^{+}} f ( x )=\displaystyle\lim _{ x \rightarrow 3^{+}}(5)=5$
$\therefore LHL \neq RHL$
Thus, $f ( x )$ is continuous everywhere except at $x =1,3$