Proving Differentiability of f(x,y)=x^2-2xy with Epsilon-Delta

[tmc]

We need to prove that f(x,y)=x^2 - 2xy is differentiable using epsilon-delta. When I do it, I just can't get rid of most of the terms.

It would be easy to prove that it's differentiable at (0,0), but differentiable at any point...it just doesn't seem to simplify.

 

[matt grime]

differentiable as in the partial derivatives exist and are continuous... that's quite easy isn't it? just write out the partials and show they are continuous

f_y= -2x

you can show that is a continuous function of x,y with epsilons and deltas, right?

 

[tmc]

Actually he stated that we should do it from the definition:

$$\[<br /> \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {x_0 ,y_0 } \right)} \frac{{\left\| {f\left( {x,y} \right) - f\left( {x_0 ,y_0 } \right) - D_f \left( {x_0 ,y_0 } \right) \cdot \left( {x - x_0 ,y - y_0 } \right)} \right\|}}{{\left\| {\left( {x - x_0 ,y - y_0 } \right)} \right\|}} = 0<br /> \]$$which is where I am having some problems.
Obviously, yes it would be easy to simply prove that it is C1, which in turn would imply differentiability...