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

The Correct Option is A, D

To solve this problem, we need to analyze the conditions given for the set \( S \) and its implications on the differentiability of functions included in it. The set \( S \) is defined as the set of all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the condition:

\(|f(x) - f(y)|^2 \le |x - y|^3 \quad \text{for all } x, y \in \mathbb{R}.\) 

We aim to determine which statements about the functions in \( S \) are true. Let's analyze each option:

1. Every function in S is differentiable.
   • This condition implies that functions \( f \) have a very small difference when \( x \) and \( y \) are close. This suggests a Lipschitz-like condition with an exponent greater than 1, which is generally sufficient to ensure differentiability. Thus, we conclude that every function in \( S \) is differentiable.
2. There exists a function \( f \in S \) such that \( f \) is differentiable, but \( f \) is not twice differentiable.
   • Examining the given condition, this statement does not hold true because the given condition is strong enough to imply smoothness up to any required degree of differentiability. This makes the function infinitely differentiable, contradicting the statement that \( f \) is not twice differentiable.
3. There exists a function \( f \in S \) such that \( f \) is twice differentiable, but \( f \) is not thrice differentiable.
   • Similar to the previous point, the condition imposed on the function ensures that it is smooth to any higher order, beyond just twice or thrice differentiable. Therefore, this statement is also false.
4. Every function in S is infinitely differentiable.
   • Given that the difference condition resembles a stronger smoothness constraint than even most Lipschitz conditions, it implies that the functions are not only smooth but indeed infinitely differentiable.

Based on the analysis, the correct statements are those that confirm the infinite differentiability of functions in \( S \). Hence, the correct answers are:

• Every function in S is differentiable.
• Every function in S is infinitely differentiable.