derivatives - How to prove this theorem about differentiability of a multivariable function?

The theorem says that:

A function $f: \mathbb{R}^2 \to \mathbb{R}$ is differentiable at $(x_0, y_0)$ if its partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ are continuous at $(x_0, y_0)$.

How do I prove this? I know that for a function to be differentiable, the condition is:

$$ \displaystyle \lim_{\lVert h,k \rVert \to 0} \dfrac{f(x_0+h, y_0+k) - f(x_0, y_0) - h\,\frac{\partial f}{\partial x}\rvert_{(x_0, y_0)} - k\,\frac{\partial f}{\partial y}\rvert_{(x_0, y_0)}}{\lVert h,k \rVert} = \lim_{\lVert h,k \rVert \to 0} \mathcal O(h,k) = 0 $$

The problem is that the definition of differentiability is using the values of partial derivatives at the point itself, and not the function. I am not understanding how I can "link" that statement to the definition of continuity of partial derivatives, which are functions themselves. I am not even getting where I could start!