Definition:Differential of Mapping/Real Function/Open Set

Definition
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 $\mathrm d f$ is the mapping $\mathrm d f : U \to \operatorname{Hom} \left({\R, \R}\right)$ defined as:
 * $\left({\mathrm d f}\right) \left({x}\right) = \mathrm d f \left({x}\right)$

where:
 * $\mathrm d f \left({x}\right)$ is the differential of $f$ at $x$
 * $\operatorname{Hom} \left({\R, \R}\right)$ is the set of all linear transformations from $\R$ to $\R$.

The differential $\mathrm d f$ can be regarded as a (real) function of two variables, defined as:
 * $\mathrm d f \left({x; h}\right) = f' \left({x}\right) h$

where $f' \left({x}\right)$ is the derivative of $f$ at $x$.