Let `f(x)={( sqrt(1+x^(2))",", x lt sqrt(3)),(sqrt(3)x-1",", sqrt(3) le x lt 4),([x]",",4 le x lt 5),(|1-x|",",x ge 5):}`, where [x] is the greatest integer `le x`. The number of points of discontinuity of f(x) in R is

Options

A. 3
B. 0
C. infinite
D. 1

Answer

D

Explanation

`LHL=lim_(x to sqrt(3))f(x)=lim_(x to sqrt(3)^(-)) =sqrt(1+(sqrt(3))^(2))=2`
`RHL=lim_(x to sqrt(3)^(+)) f(x)=(sqrt(3)*sqrt(3)) -1=2`
`rArr lim_(x to sqrt(3) ) f(x)=2`
Hence, f(x) is continuous at `x=sqrt(3)`
Now, at ` x=4,`
`LHL=lim_(x to 4^(-)) =f(x)=lim_(x to 4^(-))(sqrt(3)x-1)=4sqrt(3)-1`
`RHL=lim_(x to 4^(+)) f(x) =lim_(x to 4^(+)) [x] =4`
Hence, f(x) is discontinuous at x = 4

Now, at `x = 5, LHL =lim_(x to 5^(-)) f(x)=lim_(x to 5^(-)) [x] =4 `
`RHL=lim_(x to 5^(+)) f(x)=lim_(x to 5^(+)) |1-x| =4`
`rArr lim_(x to 5)f(x)=4`
Hence, f(x) is continuous at x = 5.