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author | immler |

Fri, 20 May 2016 22:01:42 +0200 | |

changeset 63105 | c445b0924e3a |

parent 63104 | 9505a883403c |

child 63107 | 932cf444f2fe |

uniformly continuous function extended continuously on closure

src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy | file | annotate | diff | comparison | revisions |

--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri May 20 21:21:28 2016 +0200 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri May 20 22:01:42 2016 +0200 @@ -6086,6 +6086,158 @@ qed qed +lemma uniformly_continuous_on_extension_at_closure: + fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space" + assumes uc: "uniformly_continuous_on X f" + assumes "x \<in> closure X" + obtains l where "(f \<longlongrightarrow> l) (at x within X)" +proof - + from assms obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" + by (auto simp: closure_sequential) + + from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs] + obtain l where l: "(\<lambda>n. f (xs n)) \<longlonglongrightarrow> l" + by atomize_elim (simp only: convergent_eq_cauchy) + + have "(f \<longlongrightarrow> l) (at x within X)" + proof (safe intro!: Lim_within_LIMSEQ) + fix xs' + assume "\<forall>n. xs' n \<noteq> x \<and> xs' n \<in> X" + and xs': "xs' \<longlonglongrightarrow> x" + then have "xs' n \<noteq> x" "xs' n \<in> X" for n by auto + + from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF \<open>xs' \<longlonglongrightarrow> x\<close> \<open>xs' _ \<in> X\<close>] + obtain l' where l': "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l'" + by atomize_elim (simp only: convergent_eq_cauchy) + + show "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l" + proof (rule tendstoI) + fix e::real assume "e > 0" + define e' where "e' \<equiv> e / 2" + have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def) + + have "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) l < e'" + by (simp add: \<open>0 < e'\<close> l tendstoD) + moreover + from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>e' > 0\<close>] + obtain d where d: "d > 0" "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x x' < d \<Longrightarrow> dist (f x) (f x') < e'" + by auto + have "\<forall>\<^sub>F n in sequentially. dist (xs n) (xs' n) < d" + by (auto intro!: \<open>0 < d\<close> order_tendstoD tendsto_eq_intros xs xs') + ultimately + show "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) l < e" + proof eventually_elim + case (elim n) + have "dist (f (xs' n)) l \<le> dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l" + by (metis dist_triangle dist_commute) + also have "dist (f (xs n)) (f (xs' n)) < e'" + by (auto intro!: d xs \<open>xs' _ \<in> _\<close> elim) + also note \<open>dist (f (xs n)) l < e'\<close> + also have "e' + e' = e" by (simp add: e'_def) + finally show ?case by simp + qed + qed + qed + thus ?thesis .. +qed + +lemma uniformly_continuous_on_extension_on_closure: + fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space" + assumes uc: "uniformly_continuous_on X f" + obtains g where "uniformly_continuous_on (closure X) g" "\<And>x. x \<in> X \<Longrightarrow> f x = g x" + "\<And>Y h x. X \<subseteq> Y \<Longrightarrow> Y \<subseteq> closure X \<Longrightarrow> continuous_on Y h \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> f x = h x) \<Longrightarrow> x \<in> Y \<Longrightarrow> h x = g x" +proof - + from uc have cont_f: "continuous_on X f" + by (simp add: uniformly_continuous_imp_continuous) + obtain y where y: "(f \<longlongrightarrow> y x) (at x within X)" if "x \<in> closure X" for x + apply atomize_elim + apply (rule choice) + using uniformly_continuous_on_extension_at_closure[OF assms] + by metis + let ?g = "\<lambda>x. if x \<in> X then f x else y x" + + have "uniformly_continuous_on (closure X) ?g" + unfolding uniformly_continuous_on_def + proof safe + fix e::real assume "e > 0" + define e' where "e' \<equiv> e / 3" + have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def) + from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>0 < e'\<close>] + obtain d where "d > 0" and d: "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> dist (f x') (f x) < e'" + by auto + define d' where "d' = d / 3" + have "d' > 0" using \<open>d > 0\<close> by (simp add: d'_def) + show "\<exists>d>0. \<forall>x\<in>closure X. \<forall>x'\<in>closure X. dist x' x < d \<longrightarrow> dist (?g x') (?g x) < e" + proof (safe intro!: exI[where x=d'] \<open>d' > 0\<close>) + fix x x' assume x: "x \<in> closure X" and x': "x' \<in> closure X" and dist: "dist x' x < d'" + then obtain xs xs' where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" + and xs': "xs' \<longlonglongrightarrow> x'" "\<And>n. xs' n \<in> X" + by (auto simp: closure_sequential) + have "\<forall>\<^sub>F n in sequentially. dist (xs' n) x' < d'" + and "\<forall>\<^sub>F n in sequentially. dist (xs n) x < d'" + by (auto intro!: \<open>0 < d'\<close> order_tendstoD tendsto_eq_intros xs xs') + moreover + have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x" if "x \<in> closure X" "x \<notin> X" "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" for xs x + using that not_eventuallyD + by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at) + then have "(\<lambda>x. f (xs' x)) \<longlonglongrightarrow> ?g x'" "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> ?g x" + using x x' + by (auto intro!: continuous_on_tendsto_compose[OF cont_f] simp: xs' xs) + then have "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) (?g x') < e'" + "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) (?g x) < e'" + by (auto intro!: \<open>0 < e'\<close> order_tendstoD tendsto_eq_intros) + ultimately + have "\<forall>\<^sub>F n in sequentially. dist (?g x') (?g x) < e" + proof eventually_elim + case (elim n) + have "dist (?g x') (?g x) \<le> + dist (f (xs' n)) (?g x') + dist (f (xs' n)) (f (xs n)) + dist (f (xs n)) (?g x)" + by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le) + also + { + have "dist (xs' n) (xs n) \<le> dist (xs' n) x' + dist x' x + dist (xs n) x" + by (metis add.commute add_le_cancel_left dist_triangle dist_triangle_le) + also note \<open>dist (xs' n) x' < d'\<close> + also note \<open>dist x' x < d'\<close> + also note \<open>dist (xs n) x < d'\<close> + finally have "dist (xs' n) (xs n) < d" by (simp add: d'_def) + } + with \<open>xs _ \<in> X\<close> \<open>xs' _ \<in> X\<close> have "dist (f (xs' n)) (f (xs n)) < e'" + by (rule d) + also note \<open>dist (f (xs' n)) (?g x') < e'\<close> + also note \<open>dist (f (xs n)) (?g x) < e'\<close> + finally show ?case by (simp add: e'_def) + qed + then show "dist (?g x') (?g x) < e" by simp + qed + qed + moreover have "f x = ?g x" if "x \<in> X" for x using that by simp + moreover + { + fix Y h x + assume Y: "x \<in> Y" "X \<subseteq> Y" "Y \<subseteq> closure X" and cont_h: "continuous_on Y h" + and extension: "(\<And>x. x \<in> X \<Longrightarrow> f x = h x)" + { + assume "x \<notin> X" + have "x \<in> closure X" using Y by auto + then obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" + by (auto simp: closure_sequential) + from continuous_on_tendsto_compose[OF cont_h xs(1)] xs(2) Y + have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> h x" + by (auto simp: set_mp extension) + moreover + then have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x" + using \<open>x \<notin> X\<close> not_eventuallyD xs(2) + by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at xs) + ultimately have "h x = y x" by (rule LIMSEQ_unique) + } then + have "h x = ?g x" + using extension by auto + } + ultimately show ?thesis .. +qed + + subsection\<open>Quotient maps\<close> lemma quotient_map_imp_continuous_open: