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15 (* This file was automatically generated: do not edit *********************)
17 set "baseuri" "cic:/matita/CoRN-Decl/ftc/Derivative".
19 include "CoRN_notation.ma".
21 (* $Id: Derivative.v,v 1.7 2004/04/23 10:00:58 lcf Exp $ *)
23 include "ftc/Continuity.ma".
31 We will now proceed toward the development of differential calculus.
32 To begin with, the main notion is that of derivative.
34 At this stage we will not define a notion of differentiable function,
35 mainly because the natural definition (that of being a function which
36 has some derivative) poses some technical problems; thus, we will
37 postpone that part of our work to a subsequent stage.
39 Derivative is a binary relation in the type of partial functions,
40 dependent (once again) on a compact interval with distinct
41 endpoints#. #%\footnote{%As before, we do not define pointwise
42 differentiability, mainly for coherence reasons. See Bishop [1967]
43 for a discussion on the relative little interest of that concept.%}.%
44 The reason for requiring the endpoints to be apart is mainly to be
45 able to derive the usual properties of the derivative
46 relation---namely, that any two derivatives of the same function must
49 %\begin{convention}% Let [a,b:IR] with [a [<] b] and denote by [I] the
50 interval [[a,b]]. Throughout this chapter, [F, F', G, G'] and [H]
51 will be partial functions with domains respectively [P, P', Q, Q'] and
56 inline "cic:/CoRN/ftc/Derivative/a.var".
58 inline "cic:/CoRN/ftc/Derivative/b.var".
60 inline "cic:/CoRN/ftc/Derivative/Hab'.var".
64 inline "cic:/CoRN/ftc/Derivative/Hab.con".
66 inline "cic:/CoRN/ftc/Derivative/I.con".
70 inline "cic:/CoRN/ftc/Derivative/F.var".
74 inline "cic:/CoRN/ftc/Derivative/P.con".
78 inline "cic:/CoRN/ftc/Derivative/Derivative_I.con".
85 Implicit Arguments Derivative_I [a b].
89 Section Basic_Properties.
92 (*#* **Basic Properties
95 inline "cic:/CoRN/ftc/Derivative/a.var".
97 inline "cic:/CoRN/ftc/Derivative/b.var".
99 inline "cic:/CoRN/ftc/Derivative/Hab'.var".
103 inline "cic:/CoRN/ftc/Derivative/Hab.con".
105 inline "cic:/CoRN/ftc/Derivative/I.con".
110 Like we did for equality, we begin by stating a lemma that makes proofs of derivation easier in practice.
113 inline "cic:/CoRN/ftc/Derivative/Derivative_I_char.con".
118 Derivative is a well defined relation; we will make this explicit for both arguments:
121 inline "cic:/CoRN/ftc/Derivative/F.var".
123 inline "cic:/CoRN/ftc/Derivative/G.var".
125 inline "cic:/CoRN/ftc/Derivative/H.var".
129 inline "cic:/CoRN/ftc/Derivative/P.con".
131 inline "cic:/CoRN/ftc/Derivative/Q.con".
133 inline "cic:/CoRN/ftc/Derivative/R.con".
137 inline "cic:/CoRN/ftc/Derivative/Derivative_I_wdl.con".
139 inline "cic:/CoRN/ftc/Derivative/Derivative_I_wdr.con".
143 inline "cic:/CoRN/ftc/Derivative/Derivative_I_unique_lemma.con".
148 Derivative is unique.
151 inline "cic:/CoRN/ftc/Derivative/Derivative_I_unique.con".
154 Finally, the set where we are considering the relation is included in the domain of both functions.
157 inline "cic:/CoRN/ftc/Derivative/derivative_imp_inc.con".
159 inline "cic:/CoRN/ftc/Derivative/derivative_imp_inc'.con".
162 Any function that is or has a derivative is continuous.
165 inline "cic:/CoRN/ftc/Derivative/Hab''.var".
167 inline "cic:/CoRN/ftc/Derivative/deriv_imp_contin'_I.con".
169 inline "cic:/CoRN/ftc/Derivative/deriv_imp_contin_I.con".
172 End Basic_Properties.
176 If [G] is the derivative of [F] in a given interval, then [G] is also the derivative of [F] in any smaller interval.
179 inline "cic:/CoRN/ftc/Derivative/included_imp_deriv.con".