Homotopy Type Theory differentiable function > history (Rev #6, changes)

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Definition

Valued in fields

Given an Archimedean ordered field FF, a pointwise continuous function f:FFf:F \to F is pointwise differentiable if it comes with a function D(f):FFD(f):F \to F called the derivative and such that for every positive element ϵ:F +\epsilon:F_+, there exists a positive element δ:F +\delta:F_+ such that for every element h:Fh:F such that 0<max(h,h)<δ0 \lt \max(h, -h) \lt \delta and for every element x:Fx:F,

Given a algebraic limit field FF, a subtype SFS \subseteq F, a pointwise continuous function f:C 0(S,F)f:C^0(S, F) is differentiable if the difference quotient has a limit approaching the diagonal

|f(x+h)f(x)hD(f)(x)|<ϵ\vert f(x + h) - f(x) - h D(f)(x) \vert \lt \epsilon
isDifferentiable(f):=hasLimitApproachingDiagonal(f(x)f(y)xy)isDifferentiable(f) := hasLimitApproachingDiagonal\left(\frac{f(x) - f(y)}{x - y}\right)

Let us define the type of differentiable functions D 1(S,F)D^1(S, F) on SS as the type of pointwise continuous functions that are differentiable.

D 1(S,F):= f:C 0(S,F)isDifferentiable(f)D^1(S, F) := \sum_{f:C^0(S, F)} isDifferentiable(f)

See also

Revision on June 10, 2022 at 14:11:38 by Anonymous?. See the history of this page for a list of all contributions to it.