188
III - Convergence: Continuous variables
(1.2)
(x EX) & (x < -lOP) ====} d[u,f(x)] < lO-n.
(c) Limit as x --+ a E C; this is by far the most important case. Here one
demands that for all r > 0
(1.3)
lu - f(x)1 < r for all x E X sufficiently close to a,
i.e. that there exists an r' > 0 such that, for all x E X,
(1.4)
dCa, x) < r' ====} d[u, f(x)] < r.
Then one writes
u = lim f(x).
x_a
This is the type of situation which we met in the complex case in Chap. II,
nO 19, in showing that the sum fez) of a power series is differentiable at all
points a in the disc of convergence D: in this case X = D - { a }, the function
to consider is the quotient of fez) - f(a) by z - a (not defined for z = a),
and the limit value when z tends to a is the sum f'(a) of the derived power
series.
Before proceeding, some fundamental remarks.
In the first place, one neither insists on the point a belonging to the set
X on which the function is defined, nor does one forbid this. It can happen,
in the case where the function f is defined on a set E containing a, that one
studies the behaviour of f on the set X = E - {a} obtained by deleting a
from X, or on the set X of x E E such that x > a, etc. This then has to be
specified through the notation, for example in the following way:
or
lim ;
Z_G,x>a
xEE
the second case, which arises mainly in the theories of integration and of
Fourier series, assumes that one is working in IR since inequalities are not
defined between non-real complex numbers.
In the second place, we note that the concept of convergence when x E X
tends to a point a imposes an hypothesis on a. Relative to X, the points of
IR (or of C according to the case under consideration) decompose into two
disjoint sets:
(i) First, it may happen that there exists a ball B with centre a and of
radius R > 0 such that B n X is empty, in which case one says that a is
exterior to X in IR (or q; in this case, the relation (4) is trivially satisfied for
all u and r once r' < R since then there is nothing to verify. This shows that
if one takes the definition literally, a function defined on X = (0,1) may, as
x E X tends to 2, converge indifferently to 1815, to 7r or to _10 123 ; absurd.
So we have to exclude those points exterior to X in the definition.
It is important to understand that though points exterior to X are clearly
"outside" X the converse does not hold: if X is the interval [0, 1 [ the point 1
III - Convergence: Continuous variables
(1.2)
(x EX) & (x < -lOP) ====} d[u,f(x)] < lO-n.
(c) Limit as x --+ a E C; this is by far the most important case. Here one
demands that for all r > 0
(1.3)
lu - f(x)1 < r for all x E X sufficiently close to a,
i.e. that there exists an r' > 0 such that, for all x E X,
(1.4)
dCa, x) < r' ====} d[u, f(x)] < r.
Then one writes
u = lim f(x).
x_a
This is the type of situation which we met in the complex case in Chap. II,
nO 19, in showing that the sum fez) of a power series is differentiable at all
points a in the disc of convergence D: in this case X = D - { a }, the function
to consider is the quotient of fez) - f(a) by z - a (not defined for z = a),
and the limit value when z tends to a is the sum f'(a) of the derived power
series.
Before proceeding, some fundamental remarks.
In the first place, one neither insists on the point a belonging to the set
X on which the function is defined, nor does one forbid this. It can happen,
in the case where the function f is defined on a set E containing a, that one
studies the behaviour of f on the set X = E - {a} obtained by deleting a
from X, or on the set X of x E E such that x > a, etc. This then has to be
specified through the notation, for example in the following way:
or
lim ;
Z_G,x>a
xEE
the second case, which arises mainly in the theories of integration and of
Fourier series, assumes that one is working in IR since inequalities are not
defined between non-real complex numbers.
In the second place, we note that the concept of convergence when x E X
tends to a point a imposes an hypothesis on a. Relative to X, the points of
IR (or of C according to the case under consideration) decompose into two
disjoint sets:
(i) First, it may happen that there exists a ball B with centre a and of
radius R > 0 such that B n X is empty, in which case one says that a is
exterior to X in IR (or q; in this case, the relation (4) is trivially satisfied for
all u and r once r' < R since then there is nothing to verify. This shows that
if one takes the definition literally, a function defined on X = (0,1) may, as
x E X tends to 2, converge indifferently to 1815, to 7r or to _10 123 ; absurd.
So we have to exclude those points exterior to X in the definition.
It is important to understand that though points exterior to X are clearly
"outside" X the converse does not hold: if X is the interval [0, 1 [ the point 1