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A map between open subsets of normed vector spaces that admits a linear first-order approximation at every point.
In finite-dimensional real spaces this is the standard total differentiability notion; differentiability over other scalar fields or on manifolds requires the corresponding adapted definition.
wp-differentiable-functionnlab-differentiable-mapEach relation below ends at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Require a continuous linear first-order approximation at every point of the open domain.
Feed several structures into a construction junction and impose compatibility between them.
wp-differentiable-functionnlab-differentiable-mapEach relation below starts at this concept.
This is an authored directed relation from the source endpoint to the target endpoint.
Fréchet differentiability at a point implies continuity there; a differentiable function is therefore continuous on its domain.
Record a genuine theorem implication that is not part of the target definition; these edges may point toward a weaker structure.
wp-differentiable-functionnlab-differentiable-mapThis is an authored directed relation from the source endpoint to the target endpoint.
At each point of a differentiable function, the defining first-order approximation determines a unique continuous linear derivative.
Apply a standard functorial or canonical construction whose output is not merely a reduct of the input and is not generally an equivalent presentation of the same object.
wp-derivativenlab-derivative