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17 set "baseuri" "cic:/matita/CoRN-Decl/ftc/Derivative".
19 (* $Id: Derivative.v,v 1.7 2004/04/23 10:00:58 lcf Exp $ *)
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.