If [.] denotes the greatest integer function, then f(x)=[x]^2- [x^2 ] is discontinuous at

Step-by-step solution

We have, f(x)=[x]^2- [x^2 ] aligned _ x 0⁻ f(x) & = _ h 0 [0-h]^2- [(0-h)^2 ] & =(-1)^2-0=1 aligned aligned _ x 0⁺ f(x) & = _ h 0 (0+h)^2- [(0+h)^2 ] & =0-0=0 aligned f(x) is discontinuous of x=0 Now, aligned _ x 1⁻ f(x) & = _ h 0 [1-h]^2- [(1-h)^2 ] & =0-0=0 aligned aligned _ x 1⁺ f(x) & = _ h 0 [1+h]^2- [(1+h)^2 ] & =1-1=0 aligned f(x) is continuous at x=1 . f(x) is discontinuous at all integer except 1 .