Does existence of partial derivatives implies continuity at a point $(x_0,y_0)$?

If $F: \mathbb R^2 \to \mathbb R$ and $F_x$ (partial derivative of $F$ wrt $x$) and $F_y$ exist at $(x_0,y_0)$ then the function is continuous at that point. Is this true? If not what could be a counter-example?

Solution 1:

Not true. Look at $f(x,y)= \cases{ {xy\over x^2+y^2},&$(x,y) \ne (0,0)$\cr 0,&$(x,y)=(0,0)$}$.

Here $f_x(0,0)=0=f_y(0,0)$ (as seen by applying the definitions; note, $f(0,y)$ and $f(x,0)$ are identically $0$).

But $f$ is not continuous at $(0,0)$ since the limit of $f$ as $(0,y)$ approaches $(0,0)$ is 0, but the limit as $(k,k)$ approaches 0 of $f$ is $1/2$.

See Henning's comment for a simpler, and better, example.

Solution 2:

The reason why such a statement cannot be true is, that e.g. the partial derivative $f_x$ at $(x_0,y_0)$ contains only Information of $f$ along the slice $\{(x_0+t,y_0)\mid t\in\mathbb{R}\}$.

A more interesting question therefore is, if $f$ would be continuous at $(x_0,y_0)$ if the directional derivatives $D_vf(x_0,y_0):=\left.\frac{d}{dt}\right|_{t=0}f((x_0+tv_1,y_0+tv_2)$ in every direction $v\in\mathbb{R}^2$ exist. Can you answer this?

Atajh.