[Math] Continuous but partial derivative does not exist

Question

Let $f:\mathbb{R}^2\to \mathbb{R}$ be any function. Then we know the following:

1. Differentiability implies existence of partial derivatives and continuity
2. Existence of partial derivatives does not imply continuity and hence not differnetiability.
3. Continuity does not imply differentiability.

But now my question is "Does continuity of $f$ implies existence of partial derivatives?"

Answer 1

No. Take $f(x,y)=|x|$. It is continuous, but $\frac{\partial f}{\partial x}(0,0)$ doesn't exist.