Here we define continuity and prove some basic properties of continuous functions.
*)
-inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/a.var" "Definitions_and_Basic_Results__".
+alias id "a" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/a.var".
-inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/b.var" "Definitions_and_Basic_Results__".
+alias id "b" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/b.var".
-inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hab.var" "Definitions_and_Basic_Results__".
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/Hab.var".
(* begin hide *)
(* end hide *)
-inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/F.var" "Definitions_and_Basic_Results__".
+alias id "F" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/F.var".
(* begin hide *)
Assume [F] to be continuous in [I]. Then it has a least upper bound and a greater lower bound on [I].
*)
-inline "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/contF.var" "Definitions_and_Basic_Results__".
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Definitions_and_Basic_Results/contF.var".
(* begin hide *)
We now state and prove some results about continuous functions. Assume that [I] is included in the domain of both [F] and [G].
*)
-inline "cic:/CoRN/ftc/Continuity/Local_Results/a.var" "Local_Results__".
+alias id "a" = "cic:/CoRN/ftc/Continuity/Local_Results/a.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/b.var" "Local_Results__".
+alias id "b" = "cic:/CoRN/ftc/Continuity/Local_Results/b.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/Hab.var" "Local_Results__".
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Local_Results/Hab.var".
(* begin hide *)
(* end hide *)
-inline "cic:/CoRN/ftc/Continuity/Local_Results/F.var" "Local_Results__".
+alias id "F" = "cic:/CoRN/ftc/Continuity/Local_Results/F.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/G.var" "Local_Results__".
+alias id "G" = "cic:/CoRN/ftc/Continuity/Local_Results/G.var".
(* begin hide *)
(* end hide *)
-inline "cic:/CoRN/ftc/Continuity/Local_Results/incF.var" "Local_Results__".
+alias id "incF" = "cic:/CoRN/ftc/Continuity/Local_Results/incF.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/incG.var" "Local_Results__".
+alias id "incG" = "cic:/CoRN/ftc/Continuity/Local_Results/incG.var".
(*#*
The first result does not require the function to be continuous; however, its preconditions are easily verified by continuous functions, which justifies its inclusion in this section.
Assume [F] and [G] are continuous in [I]. Then functions derived from these through algebraic operations are also continuous, provided (in the case of reciprocal and division) some extra conditions are met.
*)
-inline "cic:/CoRN/ftc/Continuity/Local_Results/contF.var" "Local_Results__".
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Local_Results/contF.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/contG.var" "Local_Results__".
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Local_Results/contG.var".
inline "cic:/CoRN/ftc/Continuity/Continuous_I_plus.con".
(* begin show *)
-inline "cic:/CoRN/ftc/Continuity/Local_Results/Hg'.var" "Local_Results__".
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg'.var".
-inline "cic:/CoRN/ftc/Continuity/Local_Results/Hg''.var" "Local_Results__".
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Local_Results/Hg''.var".
(* end show *)
Section Corolaries
*)
-inline "cic:/CoRN/ftc/Continuity/Corolaries/a.var" "Corolaries__".
+alias id "a" = "cic:/CoRN/ftc/Continuity/Corolaries/a.var".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/b.var" "Corolaries__".
+alias id "b" = "cic:/CoRN/ftc/Continuity/Corolaries/b.var".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/Hab.var" "Corolaries__".
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Corolaries/Hab.var".
(* begin hide *)
(* end hide *)
-inline "cic:/CoRN/ftc/Continuity/Corolaries/F.var" "Corolaries__".
+alias id "F" = "cic:/CoRN/ftc/Continuity/Corolaries/F.var".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/G.var" "Corolaries__".
+alias id "G" = "cic:/CoRN/ftc/Continuity/Corolaries/G.var".
(* begin hide *)
(* end hide *)
-inline "cic:/CoRN/ftc/Continuity/Corolaries/contF.var" "Corolaries__".
+alias id "contF" = "cic:/CoRN/ftc/Continuity/Corolaries/contF.var".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/contG.var" "Corolaries__".
+alias id "contG" = "cic:/CoRN/ftc/Continuity/Corolaries/contG.var".
(*#*
The corresponding properties for subtraction, division and
inline "cic:/CoRN/ftc/Continuity/Continuous_I_abs.con".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/Hg'.var" "Corolaries__".
+alias id "Hg'" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg'.var".
-inline "cic:/CoRN/ftc/Continuity/Corolaries/Hg''.var" "Corolaries__".
+alias id "Hg''" = "cic:/CoRN/ftc/Continuity/Corolaries/Hg''.var".
inline "cic:/CoRN/ftc/Continuity/Continuous_I_div.con".
We finally prove that the sum of an arbitrary family of continuous functions is still a continuous function.
*)
-inline "cic:/CoRN/ftc/Continuity/Other/Sums/a.var" "Other__Sums__".
+alias id "a" = "cic:/CoRN/ftc/Continuity/Other/Sums/a.var".
-inline "cic:/CoRN/ftc/Continuity/Other/Sums/b.var" "Other__Sums__".
+alias id "b" = "cic:/CoRN/ftc/Continuity/Other/Sums/b.var".
-inline "cic:/CoRN/ftc/Continuity/Other/Sums/Hab.var" "Other__Sums__".
+alias id "Hab" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab.var".
-inline "cic:/CoRN/ftc/Continuity/Other/Sums/Hab'.var" "Other__Sums__".
+alias id "Hab'" = "cic:/CoRN/ftc/Continuity/Other/Sums/Hab'.var".
(* begin hide *)