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I'm having problems figuring out theoretical problem on "differentiability of a function". [I hope that I spelled it right...]

Suppose that:

1. Functions f(x,y) and g(x,y) are well defined in some little domain around (0,0). (*1)

2. g(x,y) is continuous at (0,0). (*2)

3. f(x,y) is differentiable and continuous at (0,0) and f(0,0)=0. (*2)

Prove that h(x,y)=f(x,y)g(x,y) is differentiable at (0,0).

What I tried:

Let ε>0

From (*1) and (*2) I know that for every ||(x,y)||<δ_1 |g(x,y)-g(0,0)|<ε/2. (*4)

From (*1) and (*3) I know that there are a,b and u(x,y), v(x,y) so that:

f(x,y)=f(0,0)+ax+by+xu(x,u)+yv(x,u) ; u(x,y) and v(x,y) approaches 0 when (x,y) approaches (0,0). (*5)

So from (*4) and (*5) and a little of algebra I get that:

[tex]

x[g(0,0)-\epsilon/2]+y[g(0,0)-\epsilon/2]+xg(x,y)v(x,y)+yg(x,y)v(x,y) \leq h(x,y) \leq x[g(0,0)+\epsilon/2]+y[g(0,0)+\epsilon/2]+xg(x,y)v(x,y)+yg(x,y)v(x,y)

[/tex]

And at this point I have no idea what to do and I fear my approach to this problem is wrong...

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# Homework Help: Differentiability of a function

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