Subject: [HM] Bolzano and continuity 5 (of 5)
From: Walter Felscher (walter.felscher@uni-tuebingen.de)
Date: Wed Jan 12 2000 - 16:02:55 EST
Section 5.
In para.17 on p.57 it is shown that the function given by a polynomial
q(x) = a + b.x^m + c.x^n + ... + p.x^r is continuous.
Beweis. Denn wenn sich x in x+w [w: omega] ver"andert; so ist die
Aenderung, welche die Function erf"ahrt, offenbar
= b.((x+w)^m) - x^m) + c.((x+w)^n) - x^n) + ... + p.((x+w)^r - x^r) ;
eine Gr"osze, von der sich leicht darthun l"aszt, dasz sie so klein
werden k"onne, als man nur immer will, wenn man w klein genug nimmt.
Denn zu Folge des binomischen Lehrsatzes, dessen G"ultigkeit f"ur
ganze positive Exponenten wir (para.8 des binom.Lehrs.) unabh"angig
von den Untersuchungen, mit denen sich die gegenw"artige Abhandlung
besch"aftigt, dargethan haben, ist die Gr"osze:
= w . ( m.b.x^(m-1) + m.(m-1)/2 .b.w.x^(m-2) + ... + w^(m-1)
+ n.c.x^(n-1) + n.(n-1)/2 .c.w.x^(n-2) + ... + w^(r-1)
+ ...
+ r.p.x^(r-1) + r.(r-1)/2 .p.w.x^(r-2) + ... + w^(r-1) ) .
[There is an obvious, but irrelevant misprint in that for the
last members of each of the lines the factors b , c , ... , p
should be inserted.]
Die Menge der Glieder, aus welchen der in den Klammern enthaltene
Factor bestehet, ist, wie man weisz, immer nur endlich, und von dem
Werthe der Gr"oszen x und w unabh"angig; [p.58] und da diese "uberall
nur in positiver Potenz erscheinen; so ist der Werth jedes einzelnen
Gliedes, folglich auch des ganzen Ausdruckes f"ur jeden Werth von x
und w , (auch f"ur x=0), immer nur endlich. Wird aber bey einerley x ,
w verkleinert; so nehmen die Glieder, an denen w vork"ommt, ab,
w"ahrend die "ubrigen unge"andert bleiben. Bezeichnen wir also durch S
die Gr"osze, die herausk"ommt, wenn man die Werthe, die alle einzelnen
Glieder des Ausdruckes f"ur ein bestimmtes w , z.B. f"ur w_1 [w_1 :
omega index 1 ] annehmen, so zu einander addirt, als ob sie alle
einerley Vorzeichen h"atten: so ist der wirkliche Werth, den dieser
Ausdruck f"ur eben dasselbe w_1 hat, gewisz nicht >S , derjenige aber,
den er f"ur jedes kleinere w annimmt, sicher <S . Verlangt man
daher, dasz die Ver"anderung, welche die Function a + b.x^m + c.x^n +
... + p.x^r erf"ahrt, <D ausfalle; so nehme man nur ein w , das
zugleich < w_1 und auch < D/S ist: so wird w , S , und um so mehr
ein Produkt aus w in eine Gr"osze, die <S ist, <D seyn m"ussen.
Proof. If x changes into x+w then the change of the function is
= b.((x+w)^m) - x^m) + ... + p.((x+w)^r - x^r) ,
a magnitude of which it can be easily shown that it can become as
small as desired if only w is taken sufficiently small. I have
shown (para.8 of the bin.Lehrs.), independently from the
investigations of the present treatise, that the binomial
theorem holds for positive integral exponents; in consequence,
this magnitude is
= w . ( m.b.x^(m-1) + m.(m-1)/2 .b.w.x^(m-2) + ... + b.w^(m-1)
+ n.c.x^(n-1) + n.(n-1)/2 .c.w.x^(n-2) + ... + c.w^(r-1)
+ ...
+ r.p.x^(r-1) + r.(r-1)/2 .p.w.x^(r-2) + ... + p.w^(r-1) ) .
The factor in parentheses [call it T(w) in the following
comments] has a finite number of members which is independent
from the values of x and w ; as these appear always with positive
exponents, the value of each member is finite for every value of
x and w (including x=0 ), and so also the value of the entire
expression [i.e. T(w)] is finite. If x is fixed and w decreases,
then the members containing w decrease while the others remain
unchanged. If S is the expression obtained, for a certain w_1, by
adding all members of the expression [i.e. of T(w_1)] as if they
had the same sign, then the true value of the expression for w_1
[i.e. T(w_1)] certainly is not >S , while the value it accepts
for any any smaller w [i.e. T(w)] certainly is <S . Demanding
that the change of the function [i.e. q(x+w)-q(x)] be <D , take
some w such that it is both < w_1 and < D/S ; then w and S , and
much more so the product of w and a magnitude <S , must be <D
[i.e. if w < w_1 then T(w) < S , if also w < D/S then w.T(w) <
S.(D/S) = D ].
In the concluding para.18 the theorem of the title is derived from
the theorem of para.15 .
F6.
Bolzano's definition of continuity, stated in the preface, said that
given e>0 , there will hold |f(x+w)-f(x)| < e provided w is 'as
small as one wishes'. I shall observe now that wherever Bolzano uses
his definition, his use consists precisely in the verification of
given e>0 , there is d>0 such that w<d implies |f(x+w)-f(x)| < e .
These uses occur on the pages 52 and 58 .
The first was discussed in section 4. It concerned continuous
functions f , g defined in a neighbourhood of a with f(a)<g(a) of
which it was claimed that f(a+w)<g(a+w) for all w smaller than some d .
Setting A = g(a)-f(a) there holds A>0 . Given e>0 choose W such that
A-W > 0 , hence also A + W'-W for positive W' . By continuity of f
and g there is d such that w<d implies f(a+w)-f(a) <= W and
g(a+w)-g(a) <= W' Then g(a+w)-f(a+w) < e for all w<d .
The second use appeared just above in section 5. It concerned the
proof of the continuity of a polynomial q(x). Given e>0 , there was
to exhibit d>0 such that w<d implies q(x+w)-q(x) < e . To this end,
q(x+w)-q(x) was represented as w.T(w) , w_1 was chosen and S was
determined such that w < w_1 implied T(w) < S . Taking d to be the
smaller of w_1 and e/S , there followed that w<d implies q(x+w)-q(x) < e .
Thus Bolzano's notion of continuity, seemingly imprecise to today's
reader, is used by him precisely in the meaning of today's epsilon-delta
definition.
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