[Analysis] Derivative in Two Dimensions.

[tylerc1991]

Homework Statement

Let f : ℝ² -> ℝ be some function that is defined on a neighborhood of a point c in ℝ². If D₁f (the derivative of f in the direction of e₁) exists and is continuous on a neighborhood of c, and D₂f exists at c, prove that f is differentiable at c.

Homework Equations

The only sufficient condition I have for a function to be differentiable at a point is that the partial derivatives exist and are continuous at that point.

The Attempt at a Solution

If I could show that D₂f is continuous at c, then I would be done. To show that D₂f is continuous at c, I have to be sure that

|D₂f(x) - D₂f(c)| < ε

when ||x - c|| is small (do I know that D₂f(x) even exists for x ≠ c?). If f were continuous at c, then I think that I could argue that, since D₂f(x) is close to some difference quotient of f at x and D₂f(c) is close to some difference quotient of f at c (and these difference quotients can be made close to each other), then D₂f(x) is close to D₂f(c).

So now I am thinking of how to show that f is continuous. I know that if the partial derivatives are bounded on some region of c, then f is continuous at c. I think that the partial derivatives are bounded near c, but I am not totally sure. Could someone tell me if I am going in the right direction?

 

[haruspex]

f(x, y) = x² sin(1/x) for x >0, 0 otherwise.
f_y exists and is continuous in a neighbourhood of (0, 0).
f_x exists but is not continuous in a neighbourhood of (0, 0).
Does this help to narrow down the possible approaches?