What is a sufficient condition for differentiability of f:R^n --> R?

Question

I know that if all of the partial derivatives of f are continuous at a and exist in an open ball centered at a, then f is differentiable at a. But I was told differentiabilty can be ensured under less stringent conditions. Is this true? If so, can anyone give such condition, please?

Answer 1

Yes, it's true. A less rigid condition that ensures differentiability of f at a is:

One of the partial derivatives exists at a and the other n -1 are continuous at a and exist in an open ball centered at a.

So, for one of the partial derivatives, all that's required is it's existence at a (just at a, it doesn't need to exist in a neighborhood of a). And for the other n-1, it's required continuity at a and existence in an neighborhood of a. Less rigid than the condition you gave.

I guess this condition I mentioned is not very known. It's proof is in Apostol book. It's not hard, you zig-zag in the open ball where n-1 of the partial derivatives exist, in steps parallel to the axis, and apply the mean value theorem, one-dimensional case.

Answer 2

Provided it is at a constant 350, than yes the constant will provide a stronger yet more viscous reaction. In essence F an A is relative to a lesser more deliquescent tangent