Definition:Differentiable Mapping

At a Point
Let $f$ be a real function defined on an open interval $I$.

Let $\xi \in I$ be a point in $I$.

Then $f$ is differentiable at the point $\xi$ iff the limit:
 * $\displaystyle \lim_{x \to \xi} \frac {f \left({x}\right) - f \left({\xi}\right)} {x - \xi}$

exists.

This limit, if it exists, is called the derivative of $f$ at $\xi$.

On an Open Interval
Let $f$ be a real function defined on an open interval $I$.

Let $f$ be differentiable at each point of $I$.

Then $f$ is differentiable on $I$.

On a Closed Interval
Let $f$ be differentiable on $I=(a..b)$ as defined above.

If the following limit from the right exists:


 * $\displaystyle \lim_{x \to a^+} \frac {f \left({x}\right) - f \left({a}\right)} {x - a}$

as well as this limit from the left:
 * $\displaystyle \lim_{x \to b^-} \frac {f \left({x}\right) - f \left({b}\right)} {x - b}$

then $f$ is differentiable on the closed interval $[a..b]$.

On the Real Number Line
In the definition of differentiable on an interval let that interval be the real number line $\R$.

Let $f$ be differentiable at each point of $\R$.

Then $f$ is differentiable everywhere (on $\R$).

In the Complex Plane
Let $f \left({z}\right): \C \to \C$ be a single-valued continuous complex function in a domain $D \subseteq \C$.

Let $z_0 \in D$ be a point in $D$.

Then $f \left({z}\right)$ is complex-differentiable at $z_0$ iff the limit:
 * $\displaystyle \lim_{h \to 0} \frac {f \left({z_0+h}\right) - f \left({z_0}\right)} h$

exists as a finite number and is independent of how the complex increment $h$ tends to $0$.

If such a limit exists, it is called the derivative of $f$ at $z_0$.

If $f \left({z}\right)$ is complex-differentiable at every point in $D$, it is differentiable in $D$. Such a function is called analytic.

At a Point
Let $f: \mathbb X \to \R$ be a real-valued function, where $\mathbb X \subseteq \R^n$.

Let $x = ( x_1, \ldots, x_n) \in \mathbb X$.

We say that $f$ is differentiable at $x$ with respect to $x_i$ if
 * $\displaystyle \lim_{h \to 0}{\frac{f(x_1,\ldots,x_i+h,\ldots,x_n) - f(x_1,\ldots,x_i,\ldots,x_n)}{h}}$

exists and is finite.

That is, if the partial derivative $\displaystyle\frac{\partial f}{\partial x_i}$ exists.

We say that $f$ is differentiable at $x$ if $f$ is differentiable at $x$ with respect to $x_i$ for each $i=1,\ldots,n$.

See Characterization of Differentiability for an explanation of this definitions.

In a Region
Let $S \subseteq \mathbb X$.

We say that $f$ is differentiable in a region $S$ iff $f$ is differentiable at each $x$ in $S$, and write $f \in \mathcal C^1(S,\R)$.

At a Point
Let $f = (f_1,\ldots,f_m) : \mathbb X \to \R^m$ where $\mathbb X \subseteq \R^n$ be a vector valued function.

We say that $f$ is differentiable at $x \in \R^n$ if for each $j = 1,\ldots,m$, $f_j : \mathbb X \to \R$ is differentiable as a real valued function as defined above.

In a Region
Let $S \subseteq \mathbb X$.

We say that $f$ is differentiable in a region $S$ iff $f$ is differentiable at each $x$ in $S$, and write $f \in \mathcal C^1(S,\R^m)$.