Continuous partials at a point without being defined throughout a neighborhood and not differentiable there?
If I understood correctly, an example would be the following. Define $f\colon \mathbb{R}^2 \to \mathbb{R}$ as $$f(x,y) = \begin{cases}y, & \text{if }y \ge 0 \text{ or }x \le 0,\\-y,&\text{otherwise}\end{cases}$$ and take $P = (0,0)$. Then $f_x = 0$ outside $\{{0\}} \times (-\infty,0)$, $f_y(0,0) = 1$, but $f$ is not differentiable at $(0,0)$.