epsilon-delta proof

tmc

We need to prove that f(x,y)=x^2 - 2xy is differentiable using epsilon-delta. When I do it, I just cant 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 doesnt 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:

[tex]\[
\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {x_0 ,y_0 } \right)} \frac{{\left\| {f\left( {x,y} \right) - f\left( {x_0 ,y_0 } \right) - D_f \left( {x_0 ,y_0 } \right) \cdot \left( {x - x_0 ,y - y_0 } \right)} \right\|}}{{\left\| {\left( {x - x_0 ,y - y_0 } \right)} \right\|}} = 0
\]
[/tex]


which is where im having some problems.
Obviously, yes it would be easy to simply prove that it is C1, which in turn would imply differentiability...