Why is the function not differentiable at (0,0)?

[klopez]

Homework Statement

Show that the function is not differentiable at (0,0).

f(x,y) = [ (xy)/(x² + y²)^(1/2) if (x,y) =/ (0,0)

[ 0 if (x,y) = (0,0)

The Attempt at a Solution

I know that the partial derivatives at point (0,0) = 0, so I don't know why the function is not differentiable at (0,0). Is there a certain equation that will help me prove that?

Thanks

 

[HallsofIvy]

You titled this "partial differential homework" but it is important to understand that has very little to do with partial derivatives. "Differentiable" is NOT a matter of having partial derivatives.

I think it is really important that you look up the definition of "differentiable" for functions of two variables. In Calculus of one variable, we typically define the "derivative" as a limit and then say that a function is "differentiable" if and only if that limit exists. In Calculus of more than one variable, it is standard to define "differentiable" separately from just the partial derivatives.

I know several equivalent definitions of "differentiable" for two variables but I don't know which one your textbook is using: look it up please.

 

[tiny-tim]

I know that the partial derivatives at point (0,0) = 0, so I don't know why the function is not differentiable at (0,0).

Hi klopez!

It's because existence of partial derivatives ∂f/∂x and ∂f/∂y only prove differentiablity in the x and y directions.

Hint: try some other direction.