Question

Let \( f: \mathbb{R}^2 \to \mathbb{R} \) be a function. Then which of the following statements is/are TRUE?

• A. If \( f \) is differentiable at \( (0,0) \), then all directional derivatives of \( f \) exist at \( (0,0) \).
• B. If all directional derivatives of \( f \) exist at \( (0,0) \), then \( f \) is differentiable at \( (0,0) \).
• C. If all directional derivatives of \( f \) exist at \( (0,0) \), then \( f \) is continuous at \( (0,0) \).
• D. If the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist and are continuous in a disc centered at \( (0,0) \), then \( f \) is differentiable at \( (0,0) \).

Hint:

The existence and continuity of partial derivatives in a neighborhood guarantee differentiability, but the existence of directional derivatives alone does not imply differentiability.

Answer 1

The Correct Option is A, D

Step 1: Statement (A).
If \( f \) is differentiable at \( (0,0) \), then the partial derivatives of \( f \) exist at \( (0,0) \), and consequently, all directional derivatives exist at that point. This is true.
Step 2: Statement (B).
If all directional derivatives of \( f \) exist at \( (0,0) \), this does not necessarily imply that \( f \) is differentiable at \( (0,0) \). The existence of all directional derivatives does not guarantee differentiability; a counterexample can be found in non-smooth functions. Thus, this statement is false.
Step 3: Statement (C).
The existence of all directional derivatives at \( (0,0) \) does not imply continuity at that point. A function can have all directional derivatives at a point and still be discontinuous. Therefore, this statement is false.
Step 4: Statement (D).
If the partial derivatives \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) exist and are continuous in a neighborhood of \( (0,0) \), then \( f \) is differentiable at \( (0,0) \). This follows from the standard result in multivariable calculus, so this statement is true.
Step 5: Conclusion.
Thus, the correct answers are (A) and (D).