Definition:Differential/Real Function/Open Set
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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 $\rd f$ is the mapping $\rd f : U \to \operatorname{Hom} \left({\R, \R}\right)$ defined as:
- $\left({\mathrm d f}\right) \left({x}\right) = \rd f \left({x}\right)$
where:
- $\rd f \left({x}\right)$ 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 $\rd f$ can be regarded as a (real) function of two variables, defined as:
- $\rd f \left({x; h}\right) = f' \left({x}\right) h$
where $f' \left({x}\right)$ is the derivative of $f$ at $x$.