I come up with this question while working on an exercise.
Suppose that $f(x,y)$ is defined in some neighborhood of $(x_0,y_0)$, $\frac{\partial^2 f}{\partial x \partial y}$ exists and is continuous at $(x_0,y_0)$, $\partial f/\partial x$ exists. Is it true that $\partial f/\partial x$ is also continuous at $(x_0,y_0)$? I think of Leibniz rule but its hypothesis include continuity in a neighborhood and not just at a point.
I appreciate any help.