Paragraph: Let f(x)=[x]^2- [x^2 ] . Question: Consider the following statements: A. f(x) is continuous at x=0 . B. f(x) is continuous at x=1 Which of the statements given above is/are correct?

Step-by-step solution

Given f(x)=[x]^2- [x^2 ] Given, f(x)=[x]^2- [x^2 ] Let us check continuity of f(x) at x=0 aligned & LHL = _ x 0⁻ =[0- h ]^2- [(0- h )^2 ]=(-1)^2-0=1 & RHL = _ x 1⁺ =[0+ h ]^2- [(0+ h )^2 ]=0-0=0 aligned LHL RHL , so f(x) is not continous at x=0 Let us check continuity of f(x) at x=1 aligned & LHL = _ x 1⁻ =[1- h ]^2- [(1- h )^2 ]=0-0=0 & RHL = _ x 1⁺ =[1+ h ]^2- [(1+ h )^2 ]=1-1=0 aligned f ( x ) is continuous at x =1 or LHL = RHL