Theoretical question about continuity

Question

I am supposed to determine if following statement is true:

Let $f\left ( x,y \right )$ be the function of two variables defined on neighbourhood $\left ( 0,0 \right )\subseteq \mathbb{R}^{2}$. If there exist a partial derivation of function f in point $\left ( 0,0 \right )$, then the function is continous in point $\left ( 0,0 \right )$.

In my opinion yes, because it can be differentiable only if partial derivation of the function are continous, so it must be continous in point $\left ( 0,0 \right )$.

Is that true?

Answer 1

If $f(x,y)=\frac {xy} {x^{2}+y^{2}}$ for $(x,y) \neq (0,0)$ and $f(0,0)=0$ then $f$ has partial derivatives at $(0,0)$ but it is not continuous at $(0,0)$.