\begin{definition}[Definition:Differential of Mapping/Real Function/Open Set]
Let $U \subset \R$ be an open set.
Let $f : U \to \R$ be a real function.
Let $f$ be differentiable in $U$.
The '''differential''' $\d f$ is the mapping $\d f : U \to \map {\operatorname {Hom} } {\R, \R}$ defined as:
:$\map {\paren {\d f} } x = \map {\d f} x$
where:
:$\map {\d f} x$ is the differential of $f$ at $x$
:$\map {\operatorname {Hom} } {\R, \R}$ is the set of all linear transformations from $\R$ to $\R$.
The '''differential''' $\d f$ can be regarded as a (real) function of two variables, defined as:
:$\map {\d f} {x; h} = \map {f'} x h$
where $\map {f'} x$ is the derivative of $f$ at $x$.
Category:Definitions/Differentials
\end{definition}