Partially Differentiable Function

vector-calculus

Definition

Partially Differentiable Function

Let $D\subseteq {\mathbb{R}}^2$ an open set, $f:D\to \mathbb{R}$ a function and $(x_0,y_0)\in D$.

Then, $f$ is called partially differentiable with respect to $x$ at $(x_0,y_0)$, if the limit

$$\frac{\partial f}{\partial x}(x_0,y_0)=\underset{x\to x_0}{\lim }\frac{f(x,y_0)-f(x_0,y_0)}{x-x_0}$$

exists, and partially differentiable with respect to $y$, if the limit

$$\frac{\partial f}{\partial x}(x_0,y_0)=\underset{y\to y_0}{\lim }\frac{f(x_0,y)-f(x_0,y_0)}{y-y_0}$$

exists. The two limit are called the partial derivatives of $f$ with respect to $x$ and $y$, respectively.

If both partial derivative exist, $f$ is called partially differentiable. If both partial derivatives exist and both are continuous, then $f$ is called continuously partially differentiable function.