Differentiability of ${\mathbb{R}}^2$-valued functions of two variables

Let

We say that $f$ is differentiable at a point $(x_0,y_0)$ if there exists a linear map

such that

The coefficients $a_{ij}$ are the partial derivatives of the components of $f$ at $(x_0,y_0)$. More precisely, the matrix above is the Jacobian matrix

In words: the function is differentiable at $(x_0,y_0)$ if it can be well approximated near that point by a linear transformation of $(h,k)$, with an error much smaller than the distance to $(x_0,y_0)$. That is,

or, in other words,

with an error term $o(\sqrt{h^2+k^2})$.

Proposition. Relation with continuity of partial derivatives:
If all partial derivatives of the component functions $f_1,f_2$ exist in a neighborhood of $(x_0,y_0)$ and are continuous at that point, then $f$ is differentiable at $(x_0,y_0)$. (This is a sufficient but not necessary condition.)

Proposition. Directional derivatives:
Differentiability at $(x_0,y_0)$ implies the existence of all directional derivatives, and they are given by

If they satisfy Cauchy-Riemann equations then they are holomorphic functions