If `f(x)=|x-1|.([x]=[-x]),` then (where [.] represents greatest integer function)

Options

A. f(a) is continuous and differentiable at x = 1
B. f(x) is discontinuous at x = 1
C. f(x) is continuous at x = 2
D. f(x) is continuous but non-differentiable at x= 1

Answer

D

Explanation

`f(x)=|x-1|.([x]=[-x])`
`[x]-[-x]` is dicontinuous at x = 1
But g(x) = |x-1| is continuous and g(1) = 0
Hence f(x) is continuous at x = 1
Also, `f(x)={{:((1-x)(0-(-1))",",x in(0,1)),(0(-(-1))",",x=1),((x-1)(1-(-2))",",x in (1,2)):}`
`={{:(1-x",",x in (0,1]),(3(x-1)",",x in (1,2)):}`
Clearly f(x) is continuous at x = 1 and non-differentiable at x = 1 as `f'(1^(-))=-1 and f'(1^(+))=3`