Let f : R → R be a continuous function such that f x 2 = f x 3 for all x ∈ R . Consider the following statements. I. f is an odd function. II. f is an even function. III. f is differentiable everywhere. Then,

Step-by-step solution

Given function f : R ⟶ R be a continuous function such that f x 2 = f x 3 ∀ x ∈ R then f x = f x 23 [on replacing x by x 1 / 3 ] Similarly, f x = f x 23 = f x 4 / 9 = f x 827 = … = f x 23 n = f x 0 [as x tends to infinity] = f 1 ∴   f x = f 1 = constant The function f x = constant is even and differentiable everywhere.