# Sufficient condition for differentiability of a function of two variables

1. Oct 4, 2009

### AxiomOfChoice

Is there a convenient sufficient condition for knowing whether a function of two variables is differentiable? Isn't it something like if both the partial derivatives exist and are continuous, you know the derivative $\mathbf{D}f$ exists?

2. Oct 5, 2009

### HallsofIvy

Yes, that is correct. A function is differentiable at a point if and only if the partial derivatives exist and are continuous in some neighborhood of the point.

3. Oct 5, 2009

### quasar987

I'm not sure about the "only if" part.

Take for instance f(x) = x² if x is rational and =-x² otherwise. Then f is differentiable at 0 but nowhere else.

4. Oct 5, 2009

### wofsy

the derivative of a function can be discontinuous