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Let f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),...

Let `f(x)={{:(x[(1)/(x)]+x[x],if, x ne0),(0,if,x=0):}` (where [x] denotes the greatest integer function). Then the correct statement is/are

A

Limit exists for `x=-1`.

B

f(x) has a removable discontinuity at x = 1.

C

f(x) has a non removable discontinuity at x = 2.

D

f(x) is discontinuous at all positive integers.

Text Solution

Verified by Experts

The correct Answer is:
A, B, C, D
`f(1^(+))=underset(xrarr1^(+))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr1^(+))(lim)(x(0)+x(1))`
`=1`
`f(1^(-))=underset(xrarr1^(-))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr1^(-))(lim)(x(0)+x(1))`
`=1`
`f(2^(+))=underset(xrarr2^(+))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr2^(+))(lim)(x(0)+x(2))`
`=4`
`f(2^(-))=underset(xrarr2^(-))(lim)(x[(1)/(x)]+x[x])`
`=underset(xrarr2^(-))(lim)(x(0)+x(2))`
`=2`
Obviously f(x) is discontinuous at all positive integers but at x = 1 it has removable discontinuity.
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