Extending a function from a subset #

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The main definition of this file is extend_from A f where f : X → Y and A : set X. This defines a new function g : X → Y which maps any x₀ : X to the limit of f as x tends to x₀, if such a limit exists.

This is analoguous to the way dense_inducing.extend "extends" a function f : X → Z to a function g : Y → Z along a dense inducing i : X → Y.

The main theorem we prove about this definition is continuous_on_extend_from which states that, for extend_from A f to be continuous on a set B ⊆ closure A, it suffices that f converges within A at any point of B, provided that f is a function to a T₃ space.

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noncomputable def extend_from {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] (A : set X) (f : X → Y) :

X → Y

Extend a function from a set A. The resulting function g is such that at any x₀, if f converges to some y as x tends to x₀ within A, then g x₀ is defined to be one of these y. Else, g x₀ could be anything.

Equations

extend_from A f = λ (x : X), lim (nhds_within x A) f

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theorem tendsto_extend_from {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] {A : set X} {f : X → Y} {x : X} (h : ∃ (y : Y), filter.tendsto f (nhds_within x A) (nhds y)) :

filter.tendsto f (nhds_within x A) (nhds (extend_from A f x))

If f converges to some y as x tends to x₀ within A, then f tends to extend_from A f x as x tends to x₀.

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theorem extend_from_eq {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] [t2_space Y] {A : set X} {f : X → Y} {x : X} {y : Y} (hx : x ∈ closure A) (hf : filter.tendsto f (nhds_within x A) (nhds y)) :

extend_from A f x = y

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theorem extend_from_extends {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] [t2_space Y] {f : X → Y} {A : set X} (hf : continuous_on f A) (x : X) (H : x ∈ A) :

extend_from A f x = f x

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theorem continuous_on_extend_from {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] [regular_space Y] {f : X → Y} {A B : set X} (hB : B ⊆ closure A) (hf : ∀ (x : X), x ∈ B → (∃ (y : Y), filter.tendsto f (nhds_within x A) (nhds y))) :

continuous_on (extend_from A f) B

If f is a function to a T₃ space Y which has a limit within A at any point of a set B ⊆ closure A, then extend_from A f is continuous on B.

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theorem continuous_extend_from {X : Type u_1} {Y : Type u_2} [topological_space X] [topological_space Y] [regular_space Y] {f : X → Y} {A : set X} (hA : dense A) (hf : ∀ (x : X), ∃ (y : Y), filter.tendsto f (nhds_within x A) (nhds y)) :

continuous (extend_from A f)

If a function f to a T₃ space Y has a limit within a dense set A for any x, then extend_from A f is continuous.