About partial derivative and differentiability

Question

Suppose that I was given such function $f:R^2→R$ with all its partial derivative $f_x, f_y$ defined in some neighborhood of a given point $(x_0,y_0)$, also I was given that $f_x, f_y$ are continuous at such point. Does it implies that $f$ is differentiable at $(x_0,y_0)$?
Particularly note that I was only given the continuity at $(x_0, y_0)$ and not necessarily for some neighborhood. It is easy to show that if so, then $f$ would be differentiable at $(x_0,y_0)$.