Question

The function f(x) = [x] cos$\frac{1}{x - \lbrack x\rbrack}$π, (where [.] denotes the greatest integer function) is discontinuous at –

• A. All x
• B. x = $\lbrack x\rbrack - x$, n ∈ I – {1}
• C. x ∈ N
• D. x ∉ I – {0}

Answer 1

Answer: (B)

x = $\lbrack x\rbrack - x$, n ∈ I – {1}

Answer 2

Case I : when x ∈ n,

then f(x) = x cos π

f(n) = n cos (n – 1)π

f(x) = n cos(n – 1)π

f(x) = (n – 1) cos(n – 1)π

limit exist if cos (n – 1)π = 0 which is not

possible.

f(x) is discontinuous at all x ∈ I.

Case II : when x is not integer

Let = m, m is integer, then

x = = m + $\frac { 1 } { 2 }$

$\lim _ { x \rightarrow m + 1 / 2 ^ { - } } f ( x )$ = m cos (m – 1)π

= m cos mπ

limit exist only when m = 0 i.e. n = $\frac { 1 } { 2 }$

Hence discontinuous at x = $\frac { \mathrm { n } } { 2 }$ , n ∈ I – {1}.