1. The problem statement, all variables and given/known data 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. 2. Relevant 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. 3. 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?