The norm of a function is defined as being the supremum of its absolute value; that is equivalent to the following definition (which is often more convenient to use).
*)
(*#*
The norm of a function is defined as being the supremum of its absolute value; that is equivalent to the following definition (which is often more convenient to use).
We now prove some basic properties of the norm---namely that it is positive, and that it provides a least upper bound for the absolute value of its argument.
*)
(*#*
We now prove some basic properties of the norm---namely that it is positive, and that it provides a least upper bound for the absolute value of its argument.
We now state and prove some results about continuous functions. Assume that [I] is included in the domain of both [F] and [G].
*)
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/a.var.
+inline "cic:/CoRN/ftc/Continuity/a.var".
-inline cic:/CoRN/ftc/Continuity/b.var.
+inline "cic:/CoRN/ftc/Continuity/b.var".
-inline cic:/CoRN/ftc/Continuity/Hab.var.
+inline "cic:/CoRN/ftc/Continuity/Hab.var".
(* begin hide *)
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/I.con.
+inline "cic:/CoRN/ftc/Continuity/I.con".
(* end hide *)
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/F.var.
+inline "cic:/CoRN/ftc/Continuity/F.var".
-inline cic:/CoRN/ftc/Continuity/G.var.
+inline "cic:/CoRN/ftc/Continuity/G.var".
(* begin hide *)
(* begin hide *)
-inline cic:/CoRN/ftc/Continuity/P.con.
+inline "cic:/CoRN/ftc/Continuity/P.con".
-inline cic:/CoRN/ftc/Continuity/Q.con.
+inline "cic:/CoRN/ftc/Continuity/Q.con".
(* end hide *)
(* end hide *)
-inline cic:/CoRN/ftc/Continuity/incF.var.
+inline "cic:/CoRN/ftc/Continuity/incF.var".
-inline cic:/CoRN/ftc/Continuity/incG.var.
+inline "cic:/CoRN/ftc/Continuity/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.
*)
(*#*
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.
*)
(*#*
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.