- #1

Mathmos6

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Hi all, I'm looking at the following problem:

Suppose that f:[itex]\mathbb{R}^2\to\mathbb{R}[/itex] is such that [itex]\frac{\partial{f}}{\partial{x}}[/itex] is continuous in some open ball around (a,b) and [itex]\frac{\partial{f}}{\partial{y}}[/itex] exists at (a,b): show f is differentiable at (a,b).

Now I know that if both partial derivatives are continuous in a ball around (a,b) it is differentiable, but I don't know how to deal with the case where we only have continuity of one partial derivative and existence of the other - could anyone help me with this?

I've thought about it for a good hour or so now and don't seem to be getting anywhere (I know I'm meant to post what I've got so far but I literally have nothing!) so the more help you can give me the better!

Many thanks :)

Suppose that f:[itex]\mathbb{R}^2\to\mathbb{R}[/itex] is such that [itex]\frac{\partial{f}}{\partial{x}}[/itex] is continuous in some open ball around (a,b) and [itex]\frac{\partial{f}}{\partial{y}}[/itex] exists at (a,b): show f is differentiable at (a,b).

Now I know that if both partial derivatives are continuous in a ball around (a,b) it is differentiable, but I don't know how to deal with the case where we only have continuity of one partial derivative and existence of the other - could anyone help me with this?

I've thought about it for a good hour or so now and don't seem to be getting anywhere (I know I'm meant to post what I've got so far but I literally have nothing!) so the more help you can give me the better!

Many thanks :)

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