Question

The function f : R → Z defined by f(x) = [x]; where [.] denotes the greatest integer function, is ______.

Options

• Continuous at x = 2.5 but not differentiable at x = 2.5

• Not Continuous at x = 2.5 but differentiable at x = 2.5

• Not Continuous at x = 2.5 and not differentiable at x = 2.5

• Continuous as well as differentiable at x = 2.5

Answer 1

The function f : R → Z defined by f(x) = [x]; where [.] denotes the greatest integer function, is continuous as well as differentiable at x = 2.5.

Explanation:

The graph of the function f : R → Z defined by f(x) = [x] (where [.] denotes G.I.F.) is a straight line ∀ x ∈ (2.5 − h, 2.5 + h), 'h' is an infinitesimally small positive quantity. Hence, the function is continuous and differentiable at x = 2.5.