# Function with partial derivatives that exist and are both continuous at the origin but the original function is not differentiable at the origin

For the following function

$$f(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & \text{if (x,y)\neq (0,0)} \\ 0 & \text{if (x,y)=(0,0)} \\ \end{cases}$$

I know that the partials $$f_x$$ and $$f_y$$ both exist at the point $$(0,0)$$, namely $$f_x(0,0)=0$$ and $$f_y(0,0)=0$$. I also know that this function $$f(x,y)$$ is not continuous at the origin and hence it is also not differentiable at the origin.

Thus, I want to conclude that the partials must not be continuous at the origin $$(0,0)$$, via the the contrapositive of the differentiability theorem, which states that if all the partial derivatives of a function both exist and are continuous at a point, then that function is differentiable at that point.

However, the source for this problem says that this function $$f(x,y)$$ is an example of a function whose partials both exist and are continuous at $$(0,0)$$, but where the function is also not differentiable at $$(0,0)$$.

So I am confused as to whether this source contains an error or my logic surrounding the differentiability theorem is erroneous.

In review, my question is basically if the partials $$f_x$$ and $$f_y$$ for the above given $$f(x,y)$$ are actually continuous at the origin $$(0,0)$$ or whether they are discontinuous at the origin. Thanks in advance.

If it helps I will attach an image of the source of this problem:

This is the source of my confusion

• I think they wanted to say : an example of function that IS NOT continuous at $0$, but both partial derivative exist at $0$. In particular, it's not differentiable at $0$ (since not continuous)
– Surb
Aug 21, 2020 at 7:34

The source contains an error. The partial derivative w.r.t $$x$$ is $$\frac {y^{3}-x^{2}y} {(x^{2}+y^{2})^{2}}$$ which does not even have a limit along the $$y-$$ axis.