Question

Let S be the set of all functions f: $\R\rightarrow\R$ satisfying $|f(x)-f(y) |^2 \le |x - y|^3\ for\ all\ x, y \isin \R.$ Then which of the following is/are true?

• A. Every function in S is differentiable.
• B. There exists a function f ∈ S such that f is differentiable, but f is not twice differentiable.
• C. There exists a function f ∈ S such that f is twice differentiable, but f is not thrice differentiable.
• D. Every function in S is infinitely differentiable.

Answer 1

Answer: (A), (D)

Every function in S is differentiable.

Answer 2

The correct option is (A): Every function in S is differentiable. and (D): Every function in S is infinitely differentiable.