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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/metrics/ContFunctions".
19 (* $Id: ContFunctions.v,v 1.3 2004/04/23 10:01:02 lcf Exp $ *)
26 Section Continuous_functions.
29 (*#* **Continuous functions, uniformly continuous functions and Lipschitz functions
31 Let [A] and [B] be pseudo metric spaces.
35 inline cic:/CoRN/metrics/ContFunctions/A.var.
37 inline cic:/CoRN/metrics/ContFunctions/B.var.
40 We will look at some notions of continuous functions.
43 inline cic:/CoRN/metrics/ContFunctions/continuous.con.
45 inline cic:/CoRN/metrics/ContFunctions/continuous'.con.
47 inline cic:/CoRN/metrics/ContFunctions/uni_continuous.con.
49 inline cic:/CoRN/metrics/ContFunctions/uni_continuous'.con.
51 inline cic:/CoRN/metrics/ContFunctions/uni_continuous''.con.
53 inline cic:/CoRN/metrics/ContFunctions/lipschitz.con.
55 inline cic:/CoRN/metrics/ContFunctions/lipschitz'.con.
58 End Continuous_functions.
62 Implicit Arguments continuous [A B].
66 Implicit Arguments uni_continuous [A B].
70 Implicit Arguments lipschitz [A B].
74 Implicit Arguments continuous' [A B].
78 Implicit Arguments uni_continuous' [A B].
82 Implicit Arguments uni_continuous'' [A B].
86 Implicit Arguments lipschitz' [A B].
95 inline cic:/CoRN/metrics/ContFunctions/nexp_power.con.
99 inline cic:/CoRN/metrics/ContFunctions/continuous_imp_continuous'.con.
101 inline cic:/CoRN/metrics/ContFunctions/continuous'_imp_continuous.con.
103 inline cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_uni_continuous'.con.
105 inline cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous.con.
107 inline cic:/CoRN/metrics/ContFunctions/uni_continuous'_imp_uni_continuous''.con.
109 inline cic:/CoRN/metrics/ContFunctions/lipschitz_imp_lipschitz'.con.
111 inline cic:/CoRN/metrics/ContFunctions/lipschitz'_imp_lipschitz.con.
114 Every uniformly continuous function is continuous and
115 every Lipschitz function is uniformly continuous.
118 inline cic:/CoRN/metrics/ContFunctions/uni_continuous_imp_continuous.con.
120 inline cic:/CoRN/metrics/ContFunctions/lipschitz_imp_uni_continuous.con.
134 The identity function is Lipschitz.
135 Hence it is uniformly continuous and continuous.
138 inline cic:/CoRN/metrics/ContFunctions/id_is_lipschitz.con.
140 inline cic:/CoRN/metrics/ContFunctions/id_is_uni_continuous.con.
142 inline cic:/CoRN/metrics/ContFunctions/id_is_continuous.con.
152 (*#* **Constant functions
154 Let [B] and [X] be pseudo metric spaces.
159 Any constant function is Lipschitz.
160 Hence it is uniformly continuous and continuous.
163 inline cic:/CoRN/metrics/ContFunctions/B.var.
165 inline cic:/CoRN/metrics/ContFunctions/X.var.
167 inline cic:/CoRN/metrics/ContFunctions/const_fun_is_lipschitz.con.
169 inline cic:/CoRN/metrics/ContFunctions/const_fun_is_uni_continuous.con.
171 inline cic:/CoRN/metrics/ContFunctions/const_fun_is_continuous.con.
183 Let [B],[C] and [X] be pseudo metric spaces.
184 Let [f : (CSetoid_fun X B)] and
185 [g : (CSetoid_fun B C)].
190 The composition of two Lipschitz/uniformly continous/continuous functions is
191 again Lipschitz/uniformly continuous/continuous.
194 inline cic:/CoRN/metrics/ContFunctions/X.var.
196 inline cic:/CoRN/metrics/ContFunctions/B.var.
198 inline cic:/CoRN/metrics/ContFunctions/f.var.
200 inline cic:/CoRN/metrics/ContFunctions/C.var.
202 inline cic:/CoRN/metrics/ContFunctions/g.var.
204 inline cic:/CoRN/metrics/ContFunctions/comp_resp_lipschitz.con.
206 inline cic:/CoRN/metrics/ContFunctions/comp_resp_uni_continuous.con.
208 inline cic:/CoRN/metrics/ContFunctions/comp_resp_continuous.con.
221 inline cic:/CoRN/metrics/ContFunctions/MSseqLimit.con.
224 Implicit Arguments MSseqLimit [X].
227 inline cic:/CoRN/metrics/ContFunctions/MSseqLimit'.con.
230 Implicit Arguments MSseqLimit' [X].
233 inline cic:/CoRN/metrics/ContFunctions/MSseqLimit_imp_MSseqLimit'.con.
235 inline cic:/CoRN/metrics/ContFunctions/MSseqLimit'_imp_MSseqLimit.con.
237 inline cic:/CoRN/metrics/ContFunctions/seqcontinuous'.con.
240 Implicit Arguments seqcontinuous' [A B].
243 inline cic:/CoRN/metrics/ContFunctions/continuous'_imp_seqcontinuous'.con.