I have a function $f(x, y)$ which is known to be defined at some neighbourhood of $(0, 0)$ and continuous at $(0, 0).$ I need to prove that the function $g(x, y) = xyf(x, y)$ is differentiable at $(0, 0)$
According to the sufficient condition of differentiability, partial derivatives of $g(x, y)$ should be continuous in some neighbourhood of $(0, 0)$. Let's consider these derivatives: $\frac{\partial g}{\partial x} = yf(x, y) + xy\frac{\partial f}{\partial x}, \frac{\partial g}{\partial y} = xf(x, y) + xy\frac{\partial f}{\partial y}$.
How can we prove that these derivatives are continuous at some neighbourhood of $(0, 0)$, using the fact that $f(x, y)$ is continuous at $(0, 0)$? Thanks in advance for any hints!
Hint: It might be easier to prove a stronger result. Suppose $g$ is defined in a neighborhood of $(0,0)$ with $g(0,0)=0.$ If $g(x,y) =o((x^2+y^2)^{1/2})$ as $(x,y) \to (0,0),$ then $Dg(0,0)=0.$