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/- | |
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Sébastien Gouëzel | |
-/ | |
import algebra.indicator_function | |
import topology.algebra.group | |
import topology.continuous_on | |
import topology.instances.ennreal | |
/-! | |
# Semicontinuous maps | |
A function `f` from a topological space `α` to an ordered space `β` is lower semicontinuous at a | |
point `x` if, for any `y < f x`, for any `x'` close enough to `x`, one has `f x' > y`. In other | |
words, `f` can jump up, but it can not jump down. | |
Upper semicontinuous functions are defined similarly. | |
This file introduces these notions, and a basic API around them mimicking the API for continuous | |
functions. | |
## Main definitions and results | |
We introduce 4 definitions related to lower semicontinuity: | |
* `lower_semicontinuous_within_at f s x` | |
* `lower_semicontinuous_at f x` | |
* `lower_semicontinuous_on f s` | |
* `lower_semicontinuous f` | |
We build a basic API using dot notation around these notions, and we prove that | |
* constant functions are lower semicontinuous; | |
* `indicator s (λ _, y)` is lower semicontinuous when `s` is open and `0 ≤ y`, or when `s` is closed | |
and `y ≤ 0`; | |
* continuous functions are lower semicontinuous; | |
* composition with a continuous monotone functions maps lower semicontinuous functions to lower | |
semicontinuous functions. If the function is anti-monotone, it instead maps lower semicontinuous | |
functions to upper semicontinuous functions; | |
* a sum of two (or finitely many) lower semicontinuous functions is lower semicontinuous; | |
* a supremum of a family of lower semicontinuous functions is lower semicontinuous; | |
* An infinite sum of `ℝ≥0∞`-valued lower semicontinuous functions is lower semicontinuous. | |
Similar results are stated and proved for upper semicontinuity. | |
We also prove that a function is continuous if and only if it is both lower and upper | |
semicontinuous. | |
## Implementation details | |
All the nontrivial results for upper semicontinuous functions are deduced from the corresponding | |
ones for lower semicontinuous functions using `order_dual`. | |
-/ | |
open_locale topological_space big_operators ennreal | |
open set | |
variables {α : Type*} [topological_space α] {β : Type*} [preorder β] | |
{f g : α → β} {x : α} {s t : set α} {y z : β} | |
/-! ### Main definitions -/ | |
/-- A real function `f` is lower semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all | |
`x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in a general | |
preordered space, using an arbitrary `y < f x` instead of `f x - ε`. -/ | |
def lower_semicontinuous_within_at (f : α → β) (s : set α) (x : α) := | |
∀ y < f x, ∀ᶠ x' in 𝓝[s] x, y < f x' | |
/-- A real function `f` is lower semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, | |
for all `x'` close enough to `x` in `s`, then `f x'` is at least `f x - ε`. We formulate this in | |
a general preordered space, using an arbitrary `y < f x` instead of `f x - ε`.-/ | |
def lower_semicontinuous_on (f : α → β) (s : set α) := | |
∀ x ∈ s, lower_semicontinuous_within_at f s x | |
/-- A real function `f` is lower semicontinuous at `x` if, for any `ε > 0`, for all `x'` close | |
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, | |
using an arbitrary `y < f x` instead of `f x - ε`. -/ | |
def lower_semicontinuous_at (f : α → β) (x : α) := | |
∀ y < f x, ∀ᶠ x' in 𝓝 x, y < f x' | |
/-- A real function `f` is lower semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` close | |
enough to `x`, then `f x'` is at least `f x - ε`. We formulate this in a general preordered space, | |
using an arbitrary `y < f x` instead of `f x - ε`. -/ | |
def lower_semicontinuous (f : α → β) := | |
∀ x, lower_semicontinuous_at f x | |
/-- A real function `f` is upper semicontinuous at `x` within a set `s` if, for any `ε > 0`, for all | |
`x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a general | |
preordered space, using an arbitrary `y > f x` instead of `f x + ε`. -/ | |
def upper_semicontinuous_within_at (f : α → β) (s : set α) (x : α) := | |
∀ y, f x < y → ∀ᶠ x' in 𝓝[s] x, f x' < y | |
/-- A real function `f` is upper semicontinuous on a set `s` if, for any `ε > 0`, for any `x ∈ s`, | |
for all `x'` close enough to `x` in `s`, then `f x'` is at most `f x + ε`. We formulate this in a | |
general preordered space, using an arbitrary `y > f x` instead of `f x + ε`.-/ | |
def upper_semicontinuous_on (f : α → β) (s : set α) := | |
∀ x ∈ s, upper_semicontinuous_within_at f s x | |
/-- A real function `f` is upper semicontinuous at `x` if, for any `ε > 0`, for all `x'` close | |
enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered space, | |
using an arbitrary `y > f x` instead of `f x + ε`. -/ | |
def upper_semicontinuous_at (f : α → β) (x : α) := | |
∀ y, f x < y → ∀ᶠ x' in 𝓝 x, f x' < y | |
/-- A real function `f` is upper semicontinuous if, for any `ε > 0`, for any `x`, for all `x'` | |
close enough to `x`, then `f x'` is at most `f x + ε`. We formulate this in a general preordered | |
space, using an arbitrary `y > f x` instead of `f x + ε`.-/ | |
def upper_semicontinuous (f : α → β) := | |
∀ x, upper_semicontinuous_at f x | |
/-! | |
### Lower semicontinuous functions | |
-/ | |
/-! #### Basic dot notation interface for lower semicontinuity -/ | |
lemma lower_semicontinuous_within_at.mono (h : lower_semicontinuous_within_at f s x) | |
(hst : t ⊆ s) : lower_semicontinuous_within_at f t x := | |
λ y hy, filter.eventually.filter_mono (nhds_within_mono _ hst) (h y hy) | |
lemma lower_semicontinuous_within_at_univ_iff : | |
lower_semicontinuous_within_at f univ x ↔ lower_semicontinuous_at f x := | |
by simp [lower_semicontinuous_within_at, lower_semicontinuous_at, nhds_within_univ] | |
lemma lower_semicontinuous_at.lower_semicontinuous_within_at | |
(s : set α) (h : lower_semicontinuous_at f x) : lower_semicontinuous_within_at f s x := | |
λ y hy, filter.eventually.filter_mono nhds_within_le_nhds (h y hy) | |
lemma lower_semicontinuous_on.lower_semicontinuous_within_at | |
(h : lower_semicontinuous_on f s) (hx : x ∈ s) : | |
lower_semicontinuous_within_at f s x := | |
h x hx | |
lemma lower_semicontinuous_on.mono (h : lower_semicontinuous_on f s) (hst : t ⊆ s) : | |
lower_semicontinuous_on f t := | |
λ x hx, (h x (hst hx)).mono hst | |
lemma lower_semicontinuous_on_univ_iff : | |
lower_semicontinuous_on f univ ↔ lower_semicontinuous f := | |
by simp [lower_semicontinuous_on, lower_semicontinuous, lower_semicontinuous_within_at_univ_iff] | |
lemma lower_semicontinuous.lower_semicontinuous_at | |
(h : lower_semicontinuous f) (x : α) : lower_semicontinuous_at f x := | |
h x | |
lemma lower_semicontinuous.lower_semicontinuous_within_at | |
(h : lower_semicontinuous f) (s : set α) (x : α) : lower_semicontinuous_within_at f s x := | |
(h x).lower_semicontinuous_within_at s | |
lemma lower_semicontinuous.lower_semicontinuous_on | |
(h : lower_semicontinuous f) (s : set α) : lower_semicontinuous_on f s := | |
λ x hx, h.lower_semicontinuous_within_at s x | |
/-! #### Constants -/ | |
lemma lower_semicontinuous_within_at_const : | |
lower_semicontinuous_within_at (λ x, z) s x := | |
λ y hy, filter.eventually_of_forall (λ x, hy) | |
lemma lower_semicontinuous_at_const : | |
lower_semicontinuous_at (λ x, z) x := | |
λ y hy, filter.eventually_of_forall (λ x, hy) | |
lemma lower_semicontinuous_on_const : | |
lower_semicontinuous_on (λ x, z) s := | |
λ x hx, lower_semicontinuous_within_at_const | |
lemma lower_semicontinuous_const : | |
lower_semicontinuous (λ (x : α), z) := | |
λ x, lower_semicontinuous_at_const | |
/-! #### Indicators -/ | |
section | |
variables [has_zero β] | |
lemma is_open.lower_semicontinuous_indicator (hs : is_open s) (hy : 0 ≤ y) : | |
lower_semicontinuous (indicator s (λ x, y)) := | |
begin | |
assume x z hz, | |
by_cases h : x ∈ s; simp [h] at hz, | |
{ filter_upwards [hs.mem_nhds h], | |
simp [hz] { contextual := tt} }, | |
{ apply filter.eventually_of_forall (λ x', _), | |
by_cases h' : x' ∈ s; | |
simp [h', hz.trans_le hy, hz] } | |
end | |
lemma is_open.lower_semicontinuous_on_indicator (hs : is_open s) (hy : 0 ≤ y) : | |
lower_semicontinuous_on (indicator s (λ x, y)) t := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_on t | |
lemma is_open.lower_semicontinuous_at_indicator (hs : is_open s) (hy : 0 ≤ y) : | |
lower_semicontinuous_at (indicator s (λ x, y)) x := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_at x | |
lemma is_open.lower_semicontinuous_within_at_indicator (hs : is_open s) (hy : 0 ≤ y) : | |
lower_semicontinuous_within_at (indicator s (λ x, y)) t x := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_within_at t x | |
lemma is_closed.lower_semicontinuous_indicator (hs : is_closed s) (hy : y ≤ 0) : | |
lower_semicontinuous (indicator s (λ x, y)) := | |
begin | |
assume x z hz, | |
by_cases h : x ∈ s; simp [h] at hz, | |
{ apply filter.eventually_of_forall (λ x', _), | |
by_cases h' : x' ∈ s; | |
simp [h', hz, hz.trans_le hy], }, | |
{ filter_upwards [hs.is_open_compl.mem_nhds h], | |
simp [hz] { contextual := tt } } | |
end | |
lemma is_closed.lower_semicontinuous_on_indicator (hs : is_closed s) (hy : y ≤ 0) : | |
lower_semicontinuous_on (indicator s (λ x, y)) t := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_on t | |
lemma is_closed.lower_semicontinuous_at_indicator (hs : is_closed s) (hy : y ≤ 0) : | |
lower_semicontinuous_at (indicator s (λ x, y)) x := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_at x | |
lemma is_closed.lower_semicontinuous_within_at_indicator (hs : is_closed s) (hy : y ≤ 0) : | |
lower_semicontinuous_within_at (indicator s (λ x, y)) t x := | |
(hs.lower_semicontinuous_indicator hy).lower_semicontinuous_within_at t x | |
end | |
/-! #### Relationship with continuity -/ | |
theorem lower_semicontinuous_iff_is_open : | |
lower_semicontinuous f ↔ ∀ y, is_open (f ⁻¹' (Ioi y)) := | |
⟨λ H y, is_open_iff_mem_nhds.2 (λ x hx, H x y hx), λ H x y y_lt, is_open.mem_nhds (H y) y_lt⟩ | |
lemma lower_semicontinuous.is_open_preimage (hf : lower_semicontinuous f) (y : β) : | |
is_open (f ⁻¹' (Ioi y)) := | |
lower_semicontinuous_iff_is_open.1 hf y | |
section | |
variables {γ : Type*} [linear_order γ] [topological_space γ] [order_topology γ] | |
lemma continuous_within_at.lower_semicontinuous_within_at {f : α → γ} | |
(h : continuous_within_at f s x) : lower_semicontinuous_within_at f s x := | |
λ y hy, h (Ioi_mem_nhds hy) | |
lemma continuous_at.lower_semicontinuous_at {f : α → γ} | |
(h : continuous_at f x) : lower_semicontinuous_at f x := | |
λ y hy, h (Ioi_mem_nhds hy) | |
lemma continuous_on.lower_semicontinuous_on {f : α → γ} | |
(h : continuous_on f s) : lower_semicontinuous_on f s := | |
λ x hx, (h x hx).lower_semicontinuous_within_at | |
lemma continuous.lower_semicontinuous {f : α → γ} | |
(h : continuous f) : lower_semicontinuous f := | |
λ x, h.continuous_at.lower_semicontinuous_at | |
end | |
/-! ### Composition -/ | |
section | |
variables {γ : Type*} [linear_order γ] [topological_space γ] [order_topology γ] | |
variables {δ : Type*} [linear_order δ] [topological_space δ] [order_topology δ] | |
lemma continuous_at.comp_lower_semicontinuous_within_at | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_within_at f s x) | |
(gmon : monotone g) : lower_semicontinuous_within_at (g ∘ f) s x := | |
begin | |
assume y hy, | |
by_cases h : ∃ l, l < f x, | |
{ obtain ⟨z, zlt, hz⟩ : ∃ z < f x, Ioc z (f x) ⊆ g ⁻¹' (Ioi y) := | |
exists_Ioc_subset_of_mem_nhds (hg (Ioi_mem_nhds hy)) h, | |
filter_upwards [hf z zlt] with a ha, | |
calc y < g (min (f x) (f a)) : hz (by simp [zlt, ha, le_refl]) | |
... ≤ g (f a) : gmon (min_le_right _ _) }, | |
{ simp only [not_exists, not_lt] at h, | |
exact filter.eventually_of_forall (λ a, hy.trans_le (gmon (h (f a)))) } | |
end | |
lemma continuous_at.comp_lower_semicontinuous_at | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_at f x) | |
(gmon : monotone g) : lower_semicontinuous_at (g ∘ f) x := | |
begin | |
simp only [← lower_semicontinuous_within_at_univ_iff] at hf ⊢, | |
exact hg.comp_lower_semicontinuous_within_at hf gmon | |
end | |
lemma continuous.comp_lower_semicontinuous_on | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous_on f s) | |
(gmon : monotone g) : lower_semicontinuous_on (g ∘ f) s := | |
λ x hx, (hg.continuous_at).comp_lower_semicontinuous_within_at (hf x hx) gmon | |
lemma continuous.comp_lower_semicontinuous | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous f) | |
(gmon : monotone g) : lower_semicontinuous (g ∘ f) := | |
λ x, (hg.continuous_at).comp_lower_semicontinuous_at (hf x) gmon | |
lemma continuous_at.comp_lower_semicontinuous_within_at_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_within_at f s x) | |
(gmon : antitone g) : upper_semicontinuous_within_at (g ∘ f) s x := | |
@continuous_at.comp_lower_semicontinuous_within_at α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | |
lemma continuous_at.comp_lower_semicontinuous_at_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : lower_semicontinuous_at f x) | |
(gmon : antitone g) : upper_semicontinuous_at (g ∘ f) x := | |
@continuous_at.comp_lower_semicontinuous_at α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | |
lemma continuous.comp_lower_semicontinuous_on_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous_on f s) | |
(gmon : antitone g) : upper_semicontinuous_on (g ∘ f) s := | |
λ x hx, (hg.continuous_at).comp_lower_semicontinuous_within_at_antitone (hf x hx) gmon | |
lemma continuous.comp_lower_semicontinuous_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : lower_semicontinuous f) | |
(gmon : antitone g) : upper_semicontinuous (g ∘ f) := | |
λ x, (hg.continuous_at).comp_lower_semicontinuous_at_antitone (hf x) gmon | |
end | |
/-! #### Addition -/ | |
section | |
variables {ι : Type*} {γ : Type*} [linear_ordered_add_comm_monoid γ] | |
[topological_space γ] [order_topology γ] | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma lower_semicontinuous_within_at.add' {f g : α → γ} | |
(hf : lower_semicontinuous_within_at f s x) (hg : lower_semicontinuous_within_at g s x) | |
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
lower_semicontinuous_within_at (λ z, f z + g z) s x := | |
begin | |
assume y hy, | |
obtain ⟨u, v, u_open, xu, v_open, xv, h⟩ : ∃ (u v : set γ), is_open u ∧ f x ∈ u ∧ is_open v ∧ | |
g x ∈ v ∧ u ×ˢ v ⊆ {p : γ × γ | y < p.fst + p.snd} := | |
mem_nhds_prod_iff'.1 (hcont (is_open_Ioi.mem_nhds hy)), | |
by_cases hx₁ : ∃ l, l < f x, | |
{ obtain ⟨z₁, z₁lt, h₁⟩ : ∃ z₁ < f x, Ioc z₁ (f x) ⊆ u := | |
exists_Ioc_subset_of_mem_nhds (u_open.mem_nhds xu) hx₁, | |
by_cases hx₂ : ∃ l, l < g x, | |
{ obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v := | |
exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂, | |
filter_upwards [hf z₁ z₁lt, hg z₂ z₂lt] with z h₁z h₂z, | |
have A1 : min (f z) (f x) ∈ u, | |
{ by_cases H : f z ≤ f x, | |
{ simp [H], exact h₁ ⟨h₁z, H⟩ }, | |
{ simp [le_of_not_le H], exact h₁ ⟨z₁lt, le_rfl⟩, } }, | |
have A2 : min (g z) (g x) ∈ v, | |
{ by_cases H : g z ≤ g x, | |
{ simp [H], exact h₂ ⟨h₂z, H⟩ }, | |
{ simp [le_of_not_le H], exact h₂ ⟨z₂lt, le_rfl⟩, } }, | |
have : (min (f z) (f x), min (g z) (g x)) ∈ u ×ˢ v := ⟨A1, A2⟩, | |
calc y < min (f z) (f x) + min (g z) (g x) : h this | |
... ≤ f z + g z : add_le_add (min_le_left _ _) (min_le_left _ _) }, | |
{ simp only [not_exists, not_lt] at hx₂, | |
filter_upwards [hf z₁ z₁lt] with z h₁z, | |
have A1 : min (f z) (f x) ∈ u, | |
{ by_cases H : f z ≤ f x, | |
{ simp [H], exact h₁ ⟨h₁z, H⟩ }, | |
{ simp [le_of_not_le H], exact h₁ ⟨z₁lt, le_rfl⟩, } }, | |
have : (min (f z) (f x), g x) ∈ u ×ˢ v := ⟨A1, xv⟩, | |
calc y < min (f z) (f x) + g x : h this | |
... ≤ f z + g z : add_le_add (min_le_left _ _) (hx₂ (g z)) } }, | |
{ simp only [not_exists, not_lt] at hx₁, | |
by_cases hx₂ : ∃ l, l < g x, | |
{ obtain ⟨z₂, z₂lt, h₂⟩ : ∃ z₂ < g x, Ioc z₂ (g x) ⊆ v := | |
exists_Ioc_subset_of_mem_nhds (v_open.mem_nhds xv) hx₂, | |
filter_upwards [hg z₂ z₂lt] with z h₂z, | |
have A2 : min (g z) (g x) ∈ v, | |
{ by_cases H : g z ≤ g x, | |
{ simp [H], exact h₂ ⟨h₂z, H⟩ }, | |
{ simp [le_of_not_le H], exact h₂ ⟨z₂lt, le_rfl⟩, } }, | |
have : (f x, min (g z) (g x)) ∈ u ×ˢ v := ⟨xu, A2⟩, | |
calc y < f x + min (g z) (g x) : h this | |
... ≤ f z + g z : add_le_add (hx₁ (f z)) (min_le_left _ _) }, | |
{ simp only [not_exists, not_lt] at hx₁ hx₂, | |
apply filter.eventually_of_forall, | |
assume z, | |
have : (f x, g x) ∈ u ×ˢ v := ⟨xu, xv⟩, | |
calc y < f x + g x : h this | |
... ≤ f z + g z : add_le_add (hx₁ (f z)) (hx₂ (g z)) } }, | |
end | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma lower_semicontinuous_at.add' {f g : α → γ} | |
(hf : lower_semicontinuous_at f x) (hg : lower_semicontinuous_at g x) | |
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
lower_semicontinuous_at (λ z, f z + g z) x := | |
by { simp_rw [← lower_semicontinuous_within_at_univ_iff] at *, exact hf.add' hg hcont } | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma lower_semicontinuous_on.add' {f g : α → γ} | |
(hf : lower_semicontinuous_on f s) (hg : lower_semicontinuous_on g s) | |
(hcont : ∀ x ∈ s, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
lower_semicontinuous_on (λ z, f z + g z) s := | |
λ x hx, (hf x hx).add' (hg x hx) (hcont x hx) | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma lower_semicontinuous.add' {f g : α → γ} | |
(hf : lower_semicontinuous f) (hg : lower_semicontinuous g) | |
(hcont : ∀ x, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
lower_semicontinuous (λ z, f z + g z) := | |
λ x, (hf x).add' (hg x) (hcont x) | |
variable [has_continuous_add γ] | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma lower_semicontinuous_within_at.add {f g : α → γ} | |
(hf : lower_semicontinuous_within_at f s x) (hg : lower_semicontinuous_within_at g s x) : | |
lower_semicontinuous_within_at (λ z, f z + g z) s x := | |
hf.add' hg continuous_add.continuous_at | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma lower_semicontinuous_at.add {f g : α → γ} | |
(hf : lower_semicontinuous_at f x) (hg : lower_semicontinuous_at g x) : | |
lower_semicontinuous_at (λ z, f z + g z) x := | |
hf.add' hg continuous_add.continuous_at | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma lower_semicontinuous_on.add {f g : α → γ} | |
(hf : lower_semicontinuous_on f s) (hg : lower_semicontinuous_on g s) : | |
lower_semicontinuous_on (λ z, f z + g z) s := | |
hf.add' hg (λ x hx, continuous_add.continuous_at) | |
/-- The sum of two lower semicontinuous functions is lower semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma lower_semicontinuous.add {f g : α → γ} | |
(hf : lower_semicontinuous f) (hg : lower_semicontinuous g) : | |
lower_semicontinuous (λ z, f z + g z) := | |
hf.add' hg (λ x, continuous_add.continuous_at) | |
lemma lower_semicontinuous_within_at_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, lower_semicontinuous_within_at (f i) s x) : | |
lower_semicontinuous_within_at (λ z, (∑ i in a, f i z)) s x := | |
begin | |
classical, | |
induction a using finset.induction_on with i a ia IH generalizing ha, | |
{ exact lower_semicontinuous_within_at_const }, | |
{ simp only [ia, finset.sum_insert, not_false_iff], | |
exact lower_semicontinuous_within_at.add (ha _ (finset.mem_insert_self i a)) | |
(IH (λ j ja, ha j (finset.mem_insert_of_mem ja))) } | |
end | |
lemma lower_semicontinuous_at_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, lower_semicontinuous_at (f i) x) : | |
lower_semicontinuous_at (λ z, (∑ i in a, f i z)) x := | |
begin | |
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *, | |
exact lower_semicontinuous_within_at_sum ha | |
end | |
lemma lower_semicontinuous_on_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, lower_semicontinuous_on (f i) s) : | |
lower_semicontinuous_on (λ z, (∑ i in a, f i z)) s := | |
λ x hx, lower_semicontinuous_within_at_sum (λ i hi, ha i hi x hx) | |
lemma lower_semicontinuous_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, lower_semicontinuous (f i)) : | |
lower_semicontinuous (λ z, (∑ i in a, f i z)) := | |
λ x, lower_semicontinuous_at_sum (λ i hi, ha i hi x) | |
end | |
/-! #### Supremum -/ | |
section | |
variables {ι : Sort*} {δ : Type*} [complete_linear_order δ] | |
lemma lower_semicontinuous_within_at_supr {f : ι → α → δ} | |
(h : ∀ i, lower_semicontinuous_within_at (f i) s x) : | |
lower_semicontinuous_within_at (λ x', ⨆ i, f i x') s x := | |
begin | |
assume y hy, | |
rcases lt_supr_iff.1 hy with ⟨i, hi⟩, | |
filter_upwards [h i y hi] with _ hx' using lt_supr_iff.2 ⟨i, hx'⟩, | |
end | |
lemma lower_semicontinuous_within_at_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, lower_semicontinuous_within_at (f i hi) s x) : | |
lower_semicontinuous_within_at (λ x', ⨆ i hi, f i hi x') s x := | |
lower_semicontinuous_within_at_supr $ λ i, lower_semicontinuous_within_at_supr $ λ hi, h i hi | |
lemma lower_semicontinuous_at_supr {f : ι → α → δ} | |
(h : ∀ i, lower_semicontinuous_at (f i) x) : | |
lower_semicontinuous_at (λ x', ⨆ i, f i x') x := | |
begin | |
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *, | |
exact lower_semicontinuous_within_at_supr h | |
end | |
lemma lower_semicontinuous_at_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, lower_semicontinuous_at (f i hi) x) : | |
lower_semicontinuous_at (λ x', ⨆ i hi, f i hi x') x := | |
lower_semicontinuous_at_supr $ λ i, lower_semicontinuous_at_supr $ λ hi, h i hi | |
lemma lower_semicontinuous_on_supr {f : ι → α → δ} | |
(h : ∀ i, lower_semicontinuous_on (f i) s) : | |
lower_semicontinuous_on (λ x', ⨆ i, f i x') s := | |
λ x hx, lower_semicontinuous_within_at_supr (λ i, h i x hx) | |
lemma lower_semicontinuous_on_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, lower_semicontinuous_on (f i hi) s) : | |
lower_semicontinuous_on (λ x', ⨆ i hi, f i hi x') s := | |
lower_semicontinuous_on_supr $ λ i, lower_semicontinuous_on_supr $ λ hi, h i hi | |
lemma lower_semicontinuous_supr {f : ι → α → δ} | |
(h : ∀ i, lower_semicontinuous (f i)) : | |
lower_semicontinuous (λ x', ⨆ i, f i x') := | |
λ x, lower_semicontinuous_at_supr (λ i, h i x) | |
lemma lower_semicontinuous_bsupr {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, lower_semicontinuous (f i hi)) : | |
lower_semicontinuous (λ x', ⨆ i hi, f i hi x') := | |
lower_semicontinuous_supr $ λ i, lower_semicontinuous_supr $ λ hi, h i hi | |
end | |
/-! #### Infinite sums -/ | |
section | |
variables {ι : Type*} | |
lemma lower_semicontinuous_within_at_tsum {f : ι → α → ℝ≥0∞} | |
(h : ∀ i, lower_semicontinuous_within_at (f i) s x) : | |
lower_semicontinuous_within_at (λ x', ∑' i, f i x') s x := | |
begin | |
simp_rw ennreal.tsum_eq_supr_sum, | |
apply lower_semicontinuous_within_at_supr (λ b, _), | |
exact lower_semicontinuous_within_at_sum (λ i hi, h i), | |
end | |
lemma lower_semicontinuous_at_tsum {f : ι → α → ℝ≥0∞} | |
(h : ∀ i, lower_semicontinuous_at (f i) x) : | |
lower_semicontinuous_at (λ x', ∑' i, f i x') x := | |
begin | |
simp_rw [← lower_semicontinuous_within_at_univ_iff] at *, | |
exact lower_semicontinuous_within_at_tsum h | |
end | |
lemma lower_semicontinuous_on_tsum {f : ι → α → ℝ≥0∞} | |
(h : ∀ i, lower_semicontinuous_on (f i) s) : | |
lower_semicontinuous_on (λ x', ∑' i, f i x') s := | |
λ x hx, lower_semicontinuous_within_at_tsum (λ i, h i x hx) | |
lemma lower_semicontinuous_tsum {f : ι → α → ℝ≥0∞} | |
(h : ∀ i, lower_semicontinuous (f i)) : | |
lower_semicontinuous (λ x', ∑' i, f i x') := | |
λ x, lower_semicontinuous_at_tsum (λ i, h i x) | |
end | |
/-! | |
### Upper semicontinuous functions | |
-/ | |
/-! #### Basic dot notation interface for upper semicontinuity -/ | |
lemma upper_semicontinuous_within_at.mono (h : upper_semicontinuous_within_at f s x) | |
(hst : t ⊆ s) : upper_semicontinuous_within_at f t x := | |
λ y hy, filter.eventually.filter_mono (nhds_within_mono _ hst) (h y hy) | |
lemma upper_semicontinuous_within_at_univ_iff : | |
upper_semicontinuous_within_at f univ x ↔ upper_semicontinuous_at f x := | |
by simp [upper_semicontinuous_within_at, upper_semicontinuous_at, nhds_within_univ] | |
lemma upper_semicontinuous_at.upper_semicontinuous_within_at | |
(s : set α) (h : upper_semicontinuous_at f x) : upper_semicontinuous_within_at f s x := | |
λ y hy, filter.eventually.filter_mono nhds_within_le_nhds (h y hy) | |
lemma upper_semicontinuous_on.upper_semicontinuous_within_at | |
(h : upper_semicontinuous_on f s) (hx : x ∈ s) : | |
upper_semicontinuous_within_at f s x := | |
h x hx | |
lemma upper_semicontinuous_on.mono (h : upper_semicontinuous_on f s) (hst : t ⊆ s) : | |
upper_semicontinuous_on f t := | |
λ x hx, (h x (hst hx)).mono hst | |
lemma upper_semicontinuous_on_univ_iff : | |
upper_semicontinuous_on f univ ↔ upper_semicontinuous f := | |
by simp [upper_semicontinuous_on, upper_semicontinuous, upper_semicontinuous_within_at_univ_iff] | |
lemma upper_semicontinuous.upper_semicontinuous_at | |
(h : upper_semicontinuous f) (x : α) : upper_semicontinuous_at f x := | |
h x | |
lemma upper_semicontinuous.upper_semicontinuous_within_at | |
(h : upper_semicontinuous f) (s : set α) (x : α) : upper_semicontinuous_within_at f s x := | |
(h x).upper_semicontinuous_within_at s | |
lemma upper_semicontinuous.upper_semicontinuous_on | |
(h : upper_semicontinuous f) (s : set α) : upper_semicontinuous_on f s := | |
λ x hx, h.upper_semicontinuous_within_at s x | |
/-! #### Constants -/ | |
lemma upper_semicontinuous_within_at_const : | |
upper_semicontinuous_within_at (λ x, z) s x := | |
λ y hy, filter.eventually_of_forall (λ x, hy) | |
lemma upper_semicontinuous_at_const : | |
upper_semicontinuous_at (λ x, z) x := | |
λ y hy, filter.eventually_of_forall (λ x, hy) | |
lemma upper_semicontinuous_on_const : | |
upper_semicontinuous_on (λ x, z) s := | |
λ x hx, upper_semicontinuous_within_at_const | |
lemma upper_semicontinuous_const : | |
upper_semicontinuous (λ (x : α), z) := | |
λ x, upper_semicontinuous_at_const | |
/-! #### Indicators -/ | |
section | |
variables [has_zero β] | |
lemma is_open.upper_semicontinuous_indicator (hs : is_open s) (hy : y ≤ 0) : | |
upper_semicontinuous (indicator s (λ x, y)) := | |
@is_open.lower_semicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy | |
lemma is_open.upper_semicontinuous_on_indicator (hs : is_open s) (hy : y ≤ 0) : | |
upper_semicontinuous_on (indicator s (λ x, y)) t := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_on t | |
lemma is_open.upper_semicontinuous_at_indicator (hs : is_open s) (hy : y ≤ 0) : | |
upper_semicontinuous_at (indicator s (λ x, y)) x := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_at x | |
lemma is_open.upper_semicontinuous_within_at_indicator (hs : is_open s) (hy : y ≤ 0) : | |
upper_semicontinuous_within_at (indicator s (λ x, y)) t x := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_within_at t x | |
lemma is_closed.upper_semicontinuous_indicator (hs : is_closed s) (hy : 0 ≤ y) : | |
upper_semicontinuous (indicator s (λ x, y)) := | |
@is_closed.lower_semicontinuous_indicator α _ βᵒᵈ _ s y _ hs hy | |
lemma is_closed.upper_semicontinuous_on_indicator (hs : is_closed s) (hy : 0 ≤ y) : | |
upper_semicontinuous_on (indicator s (λ x, y)) t := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_on t | |
lemma is_closed.upper_semicontinuous_at_indicator (hs : is_closed s) (hy : 0 ≤ y) : | |
upper_semicontinuous_at (indicator s (λ x, y)) x := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_at x | |
lemma is_closed.upper_semicontinuous_within_at_indicator (hs : is_closed s) (hy : 0 ≤ y) : | |
upper_semicontinuous_within_at (indicator s (λ x, y)) t x := | |
(hs.upper_semicontinuous_indicator hy).upper_semicontinuous_within_at t x | |
end | |
/-! #### Relationship with continuity -/ | |
theorem upper_semicontinuous_iff_is_open : | |
upper_semicontinuous f ↔ ∀ y, is_open (f ⁻¹' (Iio y)) := | |
⟨λ H y, is_open_iff_mem_nhds.2 (λ x hx, H x y hx), λ H x y y_lt, is_open.mem_nhds (H y) y_lt⟩ | |
lemma upper_semicontinuous.is_open_preimage (hf : upper_semicontinuous f) (y : β) : | |
is_open (f ⁻¹' (Iio y)) := | |
upper_semicontinuous_iff_is_open.1 hf y | |
section | |
variables {γ : Type*} [linear_order γ] [topological_space γ] [order_topology γ] | |
lemma continuous_within_at.upper_semicontinuous_within_at {f : α → γ} | |
(h : continuous_within_at f s x) : upper_semicontinuous_within_at f s x := | |
λ y hy, h (Iio_mem_nhds hy) | |
lemma continuous_at.upper_semicontinuous_at {f : α → γ} | |
(h : continuous_at f x) : upper_semicontinuous_at f x := | |
λ y hy, h (Iio_mem_nhds hy) | |
lemma continuous_on.upper_semicontinuous_on {f : α → γ} | |
(h : continuous_on f s) : upper_semicontinuous_on f s := | |
λ x hx, (h x hx).upper_semicontinuous_within_at | |
lemma continuous.upper_semicontinuous {f : α → γ} | |
(h : continuous f) : upper_semicontinuous f := | |
λ x, h.continuous_at.upper_semicontinuous_at | |
end | |
/-! ### Composition -/ | |
section | |
variables {γ : Type*} [linear_order γ] [topological_space γ] [order_topology γ] | |
variables {δ : Type*} [linear_order δ] [topological_space δ] [order_topology δ] | |
lemma continuous_at.comp_upper_semicontinuous_within_at | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_within_at f s x) | |
(gmon : monotone g) : upper_semicontinuous_within_at (g ∘ f) s x := | |
@continuous_at.comp_lower_semicontinuous_within_at α _ x s γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual | |
lemma continuous_at.comp_upper_semicontinuous_at | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_at f x) | |
(gmon : monotone g) : upper_semicontinuous_at (g ∘ f) x := | |
@continuous_at.comp_lower_semicontinuous_at α _ x γᵒᵈ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon.dual | |
lemma continuous.comp_upper_semicontinuous_on | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous_on f s) | |
(gmon : monotone g) : upper_semicontinuous_on (g ∘ f) s := | |
λ x hx, (hg.continuous_at).comp_upper_semicontinuous_within_at (hf x hx) gmon | |
lemma continuous.comp_upper_semicontinuous | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous f) | |
(gmon : monotone g) : upper_semicontinuous (g ∘ f) := | |
λ x, (hg.continuous_at).comp_upper_semicontinuous_at (hf x) gmon | |
lemma continuous_at.comp_upper_semicontinuous_within_at_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_within_at f s x) | |
(gmon : antitone g) : lower_semicontinuous_within_at (g ∘ f) s x := | |
@continuous_at.comp_upper_semicontinuous_within_at α _ x s γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | |
lemma continuous_at.comp_upper_semicontinuous_at_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous_at g (f x)) (hf : upper_semicontinuous_at f x) | |
(gmon : antitone g) : lower_semicontinuous_at (g ∘ f) x := | |
@continuous_at.comp_upper_semicontinuous_at α _ x γ _ _ _ δᵒᵈ _ _ _ g f hg hf gmon | |
lemma continuous.comp_upper_semicontinuous_on_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous_on f s) | |
(gmon : antitone g) : lower_semicontinuous_on (g ∘ f) s := | |
λ x hx, (hg.continuous_at).comp_upper_semicontinuous_within_at_antitone (hf x hx) gmon | |
lemma continuous.comp_upper_semicontinuous_antitone | |
{g : γ → δ} {f : α → γ} (hg : continuous g) (hf : upper_semicontinuous f) | |
(gmon : antitone g) : lower_semicontinuous (g ∘ f) := | |
λ x, (hg.continuous_at).comp_upper_semicontinuous_at_antitone (hf x) gmon | |
end | |
/-! #### Addition -/ | |
section | |
variables {ι : Type*} {γ : Type*} [linear_ordered_add_comm_monoid γ] | |
[topological_space γ] [order_topology γ] | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma upper_semicontinuous_within_at.add' {f g : α → γ} | |
(hf : upper_semicontinuous_within_at f s x) (hg : upper_semicontinuous_within_at g s x) | |
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
upper_semicontinuous_within_at (λ z, f z + g z) s x := | |
@lower_semicontinuous_within_at.add' α _ x s γᵒᵈ _ _ _ _ _ hf hg hcont | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma upper_semicontinuous_at.add' {f g : α → γ} | |
(hf : upper_semicontinuous_at f x) (hg : upper_semicontinuous_at g x) | |
(hcont : continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
upper_semicontinuous_at (λ z, f z + g z) x := | |
by { simp_rw [← upper_semicontinuous_within_at_univ_iff] at *, exact hf.add' hg hcont } | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma upper_semicontinuous_on.add' {f g : α → γ} | |
(hf : upper_semicontinuous_on f s) (hg : upper_semicontinuous_on g s) | |
(hcont : ∀ x ∈ s, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
upper_semicontinuous_on (λ z, f z + g z) s := | |
λ x hx, (hf x hx).add' (hg x hx) (hcont x hx) | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with an | |
explicit continuity assumption on addition, for application to `ereal`. The unprimed version of | |
the lemma uses `[has_continuous_add]`. -/ | |
lemma upper_semicontinuous.add' {f g : α → γ} | |
(hf : upper_semicontinuous f) (hg : upper_semicontinuous g) | |
(hcont : ∀ x, continuous_at (λ (p : γ × γ), p.1 + p.2) (f x, g x)) : | |
upper_semicontinuous (λ z, f z + g z) := | |
λ x, (hf x).add' (hg x) (hcont x) | |
variable [has_continuous_add γ] | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma upper_semicontinuous_within_at.add {f g : α → γ} | |
(hf : upper_semicontinuous_within_at f s x) (hg : upper_semicontinuous_within_at g s x) : | |
upper_semicontinuous_within_at (λ z, f z + g z) s x := | |
hf.add' hg continuous_add.continuous_at | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma upper_semicontinuous_at.add {f g : α → γ} | |
(hf : upper_semicontinuous_at f x) (hg : upper_semicontinuous_at g x) : | |
upper_semicontinuous_at (λ z, f z + g z) x := | |
hf.add' hg continuous_add.continuous_at | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma upper_semicontinuous_on.add {f g : α → γ} | |
(hf : upper_semicontinuous_on f s) (hg : upper_semicontinuous_on g s) : | |
upper_semicontinuous_on (λ z, f z + g z) s := | |
hf.add' hg (λ x hx, continuous_add.continuous_at) | |
/-- The sum of two upper semicontinuous functions is upper semicontinuous. Formulated with | |
`[has_continuous_add]`. The primed version of the lemma uses an explicit continuity assumption on | |
addition, for application to `ereal`. -/ | |
lemma upper_semicontinuous.add {f g : α → γ} | |
(hf : upper_semicontinuous f) (hg : upper_semicontinuous g) : | |
upper_semicontinuous (λ z, f z + g z) := | |
hf.add' hg (λ x, continuous_add.continuous_at) | |
lemma upper_semicontinuous_within_at_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, upper_semicontinuous_within_at (f i) s x) : | |
upper_semicontinuous_within_at (λ z, (∑ i in a, f i z)) s x := | |
@lower_semicontinuous_within_at_sum α _ x s ι γᵒᵈ _ _ _ _ f a ha | |
lemma upper_semicontinuous_at_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, upper_semicontinuous_at (f i) x) : | |
upper_semicontinuous_at (λ z, (∑ i in a, f i z)) x := | |
begin | |
simp_rw [← upper_semicontinuous_within_at_univ_iff] at *, | |
exact upper_semicontinuous_within_at_sum ha | |
end | |
lemma upper_semicontinuous_on_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, upper_semicontinuous_on (f i) s) : | |
upper_semicontinuous_on (λ z, (∑ i in a, f i z)) s := | |
λ x hx, upper_semicontinuous_within_at_sum (λ i hi, ha i hi x hx) | |
lemma upper_semicontinuous_sum {f : ι → α → γ} {a : finset ι} | |
(ha : ∀ i ∈ a, upper_semicontinuous (f i)) : | |
upper_semicontinuous (λ z, (∑ i in a, f i z)) := | |
λ x, upper_semicontinuous_at_sum (λ i hi, ha i hi x) | |
end | |
/-! #### Infimum -/ | |
section | |
variables {ι : Sort*} {δ : Type*} [complete_linear_order δ] | |
lemma upper_semicontinuous_within_at_infi {f : ι → α → δ} | |
(h : ∀ i, upper_semicontinuous_within_at (f i) s x) : | |
upper_semicontinuous_within_at (λ x', ⨅ i, f i x') s x := | |
@lower_semicontinuous_within_at_supr α _ x s ι δᵒᵈ _ f h | |
lemma upper_semicontinuous_within_at_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, upper_semicontinuous_within_at (f i hi) s x) : | |
upper_semicontinuous_within_at (λ x', ⨅ i hi, f i hi x') s x := | |
upper_semicontinuous_within_at_infi $ λ i, upper_semicontinuous_within_at_infi $ λ hi, h i hi | |
lemma upper_semicontinuous_at_infi {f : ι → α → δ} | |
(h : ∀ i, upper_semicontinuous_at (f i) x) : | |
upper_semicontinuous_at (λ x', ⨅ i, f i x') x := | |
@lower_semicontinuous_at_supr α _ x ι δᵒᵈ _ f h | |
lemma upper_semicontinuous_at_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, upper_semicontinuous_at (f i hi) x) : | |
upper_semicontinuous_at (λ x', ⨅ i hi, f i hi x') x := | |
upper_semicontinuous_at_infi $ λ i, upper_semicontinuous_at_infi $ λ hi, h i hi | |
lemma upper_semicontinuous_on_infi {f : ι → α → δ} | |
(h : ∀ i, upper_semicontinuous_on (f i) s) : | |
upper_semicontinuous_on (λ x', ⨅ i, f i x') s := | |
λ x hx, upper_semicontinuous_within_at_infi (λ i, h i x hx) | |
lemma upper_semicontinuous_on_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, upper_semicontinuous_on (f i hi) s) : | |
upper_semicontinuous_on (λ x', ⨅ i hi, f i hi x') s := | |
upper_semicontinuous_on_infi $ λ i, upper_semicontinuous_on_infi $ λ hi, h i hi | |
lemma upper_semicontinuous_infi {f : ι → α → δ} | |
(h : ∀ i, upper_semicontinuous (f i)) : | |
upper_semicontinuous (λ x', ⨅ i, f i x') := | |
λ x, upper_semicontinuous_at_infi (λ i, h i x) | |
lemma upper_semicontinuous_binfi {p : ι → Prop} {f : Π i (h : p i), α → δ} | |
(h : ∀ i hi, upper_semicontinuous (f i hi)) : | |
upper_semicontinuous (λ x', ⨅ i hi, f i hi x') := | |
upper_semicontinuous_infi $ λ i, upper_semicontinuous_infi $ λ hi, h i hi | |
end | |
section | |
variables {γ : Type*} [linear_order γ] [topological_space γ] [order_topology γ] | |
lemma continuous_within_at_iff_lower_upper_semicontinuous_within_at {f : α → γ} : | |
continuous_within_at f s x ↔ | |
lower_semicontinuous_within_at f s x ∧ upper_semicontinuous_within_at f s x:= | |
begin | |
refine ⟨λ h, ⟨h.lower_semicontinuous_within_at, h.upper_semicontinuous_within_at⟩, _⟩, | |
rintros ⟨h₁, h₂⟩, | |
assume v hv, | |
simp only [filter.mem_map], | |
by_cases Hl : ∃ l, l < f x, | |
{ rcases exists_Ioc_subset_of_mem_nhds hv Hl with ⟨l, lfx, hl⟩, | |
by_cases Hu : ∃ u, f x < u, | |
{ rcases exists_Ico_subset_of_mem_nhds hv Hu with ⟨u, fxu, hu⟩, | |
filter_upwards [h₁ l lfx, h₂ u fxu] with a lfa fau, | |
cases le_or_gt (f a) (f x) with h h, | |
{ exact hl ⟨lfa, h⟩ }, | |
{ exact hu ⟨le_of_lt h, fau⟩ } }, | |
{ simp only [not_exists, not_lt] at Hu, | |
filter_upwards [h₁ l lfx] with a lfa using hl ⟨lfa, Hu (f a)⟩, }, }, | |
{ simp only [not_exists, not_lt] at Hl, | |
by_cases Hu : ∃ u, f x < u, | |
{ rcases exists_Ico_subset_of_mem_nhds hv Hu with ⟨u, fxu, hu⟩, | |
filter_upwards [h₂ u fxu] with a lfa, | |
apply hu, | |
exact ⟨Hl (f a), lfa⟩ }, | |
{ simp only [not_exists, not_lt] at Hu, | |
apply filter.eventually_of_forall, | |
assume a, | |
have : f a = f x := le_antisymm (Hu _) (Hl _), | |
rw this, | |
exact mem_of_mem_nhds hv } } | |
end | |
lemma continuous_at_iff_lower_upper_semicontinuous_at {f : α → γ} : | |
continuous_at f x ↔ (lower_semicontinuous_at f x ∧ upper_semicontinuous_at f x) := | |
by simp_rw [← continuous_within_at_univ, ← lower_semicontinuous_within_at_univ_iff, | |
← upper_semicontinuous_within_at_univ_iff, | |
continuous_within_at_iff_lower_upper_semicontinuous_within_at] | |
lemma continuous_on_iff_lower_upper_semicontinuous_on {f : α → γ} : | |
continuous_on f s ↔ (lower_semicontinuous_on f s ∧ upper_semicontinuous_on f s) := | |
begin | |
simp only [continuous_on, continuous_within_at_iff_lower_upper_semicontinuous_within_at], | |
exact ⟨λ H, ⟨λ x hx, (H x hx).1, λ x hx, (H x hx).2⟩, λ H x hx, ⟨H.1 x hx, H.2 x hx⟩⟩ | |
end | |
lemma continuous_iff_lower_upper_semicontinuous {f : α → γ} : | |
continuous f ↔ (lower_semicontinuous f ∧ upper_semicontinuous f) := | |
by simp_rw [continuous_iff_continuous_on_univ, continuous_on_iff_lower_upper_semicontinuous_on, | |
lower_semicontinuous_on_univ_iff, upper_semicontinuous_on_univ_iff] | |
end | |