The question is as follows:
Let $F:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function given by $$F(x,y)=\bigg\{\begin{array}{cc} x^3\sin(1/x)+y^2 & x\neq 0\\ y^2&x=0. \end{array}$$ Show that $F$ is differentiable at the point $(0,0)$.
The piecewise nature of this function is throwing me off. I know that if all the partial derivatives are continuous in a neighborhood of the point, then the function is differentiable there. But $\frac{\partial}{\partial x}(x^3\sin(1/x)+y^2)=x(3x\sin(1/x)-\cos(1/x))$ is not continuous near $(0,0)$. So my inclination is to use the limit definition of derivative. However, $\lim_{(h,k)\rightarrow(0,0)}\frac{h^3\sin(1/h)+k^2+h-k}{\sqrt{h^2+k^2}}$ does not exist. Am I able to completely ignore the first piece since the function is not defined by it at $(0,0)$?