Thread: Proving a claim regarding differentiability

1. Proving a claim regarding differentiability

Let F(x,y,z) be a function which is defined in the point M_0(x_0,y_0,z_0) and around it and the following conditions are satisfied:

1. F(x_0,y_0,z_0)=0
2. F has continuous partial derivatives in M_0 and around it
3. F'_z(x_0,y_0,z_0)=0
4. gradF at (x_0,y_0,z_0) != 0
5. It is known that there is a function f(x,y) so that F(x,y,f(x,y)) =0 in M_0 and around it

Prove that f(x,y) is not differentiable in (x_0, y_0)

2. So, it is the "f" in (5) that you want to prove differentiable and not F?

Do you recall the definition of "differentiable" for a function of two variables?

f(x, y) is differentiable at $\displaystyle (x_0, y_0)$ if and only if there exist a linear function L(x,y)= ax+ by and a function $\displaystyle \epsilon(x,y)$ such that
1) $\displaystyle f(x, y)= F(x_0, y_0)+ a(x- x_0)+ b(y- y_0)+ \epsilon(x,y)$ and
2) $\displaystyle \lim_{(x,y)to (x_0,y_0)} \frac{\epsilon(x,y)}{\sqrt{(x-x_0)^2+ (y-y_0)^2}}= 0$

The demominator in (2) can be replaced with any reasonable measure of "distance" from $\displaystyle (x_0, y_0)$

3. I know the definition but I'm skeptical whether it helps here.

I should probably use the chain rule somehow. Can't really point my finger how though.

Help?