f x = lim n → ∞ ⁡ cos 2 n ⁡ π x 2 + x (where, ⋅ denotes the greatest integer function and n ∈ N ) is

Step-by-step solution

f x = lim n → ∞ ⁡ cos 2 ⁡ π x 2 n + x = x cos 2 ⁡ π x 2 ∈ 0 , 1 1 + x cos 2 ⁡ π x 2 = 1 = x x 2 ∉ I 1 + x   x 2 ∈ I For x = 1 f 1 - = 1 - = 0 ,   f 1 + = 1 + = 1 ,   f 1 = 1 + 1 = 2 ∴ f 1 + ≠ f 1 - ∴ f x is discontinuous at x = 1 For x = 0 f 0 - = 0 - = - 1 ,   f 0 + = 0 + = 0 ,   f 0 = 1 + 0 = 1 ∴ f 0 + ≠ f 0 - ∴ f x is discontinuous at x = 0