Partial derivative implies continuity

Question

Let $f:\mathbb{R}^2\to \mathbb{R}$. It is true that if $f_x(0,0)=f_y(0,0)$ then $f(x,y)$ is continue at the point (0,0)?

Answer 1

Nope! Take, for example: $$f(x, y) = \begin{cases} 0 & \text{if }xy = 0 \\ 1 & \text{if } xy \neq 0.\end{cases}$$ Then $f$ is $0$ along the $x$ and $y$ axes, so $f_x(0, 0) = f_y(0, 0) = 0$, but $f$ is not continuous at $(0, 0)$ as the points $(t, t)$ for $t > 0$ approaches $(0, 0)$ as $t \to 0^+$, and $f(t, t) = 1$ for all such $t$.