Homotopy Type Theory
differentiable function > history (Rev #6, changes)
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Definition
Valued in fields Given an Archimedean ordered field F F , a pointwise continuous function f : F → F f:F \to F is pointwise differentiable if it comes with a function D ( f ) : F → F D(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 : F h:F such that 0 < max ( h , − h ) < δ 0 \lt \max(h, -h) \lt \delta and for every element x : F x:F ,
Given a algebraic limit field F F , a subtype S ⊆ F S \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 ) − h D ( f ) ( x ) | < ϵ \vert f(x + h) - f(x) - h D(f)(x) \vert \lt \epsilon
isDifferentiable ( f ) : = hasLimitApproachingDiagonal ( f ( x ) − f ( y ) x − y ) 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 S S 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
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