The number of points of discontinuity of f ( x ) = ( sgn ( x ) ) ( sgn ( - x ) ) in R is i) 0 2) The number of points of continuity of f ( x ) = { e | x | - 1 , x is rational 1 1 + x 2 , x is irrational in R is ii) 1

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Q.
Match the following

Column – I Column - II

1) The number of points of discontinuity of $f (x) = (sgn (x)) (sgn (- x))$ in $R$ is i) 0

2) The number of points of continuity of $f$ $(x)$ $=$ ${\begin{matrix} e^{| x |} - 1, & x is rational \\ \frac{1}{1 + x^{2}}, & x is irrational \end{matrix}$ in $R$ is ii) 1

3) The number of points of discontinuity of $g (x) = \lim_{n \to \infty} \frac{(f {(x))}^{n} - 1}{(f {(x))}^{n} + 1} in (0, \infty)$ where $f (x) = (x - 1) (x - 2) (x - 3) + 1$ is iii) 2

4) The number of points of discontinuity of $f (x) = \max ([x], | x |)$ in $R$ is, where $[.]$ represents the GIF iv) 3

Which of the following match is correct

Solution

1 1–iii, 2–iv, 3–i, 4–ii

2 1–ii, 2–iii, 3–iv, 4–i

3 1–iii, 2–i, 3–iv, 4–ii

4 1–iv, 2–i, 3–ii, 4–iii

Ans.
(2)

conceptual

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	Column – I		Column - II
1)	The number of points of discontinuity of $f (x) = (sgn (x)) (sgn (- x))$ in $R$ is	i)	0
2)	The number of points of continuity of $f$ $(x)$ $=$ ${\begin{matrix} e^{\| x \|} - 1, & x is rational \\ \frac{1}{1 + x^{2}}, & x is irrational \end{matrix}$ in $R$ is	ii)	1
3)	The number of points of discontinuity of $g (x) = \lim_{n \to \infty} \frac{(f {(x))}^{n} - 1}{(f {(x))}^{n} + 1} in (0, \infty)$ where $f (x) = (x - 1) (x - 2) (x - 3) + 1$ is	iii)	2
4)	The number of points of discontinuity of $f (x) = \max ([x], \| x \|)$ in $R$ is, where $[.]$ represents the GIF	iv)	3

Solution

1 1–iii, 2–iv, 3–i, 4–ii

2 1–ii, 2–iii, 3–iv, 4–i

3 1–iii, 2–i, 3–iv, 4–ii

4 1–iv, 2–i, 3–ii, 4–iii

Ans. (2) conceptual

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Ans.
(2)

conceptual