64
II - Convergence: Discrete variables
inequality III zl < r is satisfied as soon as Izl > l/r. Similarly, x 2 tends to 4
when x tends to 2, since on the one hand
Ix 2 - 41 = Ix - 21·lx + 21 < 51x - 21 for Ix - 21 ~ 1,
while on the other hand, 51x - 21 < r for Ix - 21 < r 15; so Ix 2 - 41 < r once
Ix - 21 < r' = min(1,rI5).
We shall hardly ever, in this chapter, use the concept of limit except for
the case of sequences, Le. of functions where the independent variable takes
only integer values. The general case which we have just mentioned will be
covered in detail in Chap. III. Since we shall nevertheless need to speak
occasionally of continuity and of differentiability in this chapter, let us now
give the definitions of these two fundamental properties.
A scalar (Le. complex-valued) function f defined on a set X C C is said
to be continuous at a point a of X if
limf(x) = f(a) when x ~ a.
This means that for all r > 0 there exists an r' > 0 such that
(4.4)
{(x E X) & (Ix - al < r'n =? If(x) - f(a)1 < r
or again that, for all r > 0, f(x) is constant to within r on a neighbourhood
of a in X or again, in decimal language, that if one wants to calculate f(a)
to within lO-n it suffices to calculate f(x) for any x E X having sufficiently
many decimal places in common with a, for example the number obtained
by replacing all the digits of a of sufficiently high rank by 0; this is what all
the practitioners of numerical analysis have always done, and this is what
all computers do. The calculation above, showing that x 2 tends to 4 when x
tends to 2, expresses the continuity of the function x r---+ x 2 at x = 2.
More generally, let us show, as a useful exercise, that the functions x n , or,
as clearly amounts to the same, f(x) = x[n] = xnln!, are continuous on C. It
suffices to show - the other formulation of continuity - that, for x given, the
difference If(x + h) - f(x)1 is < r for Ihl sufficiently small. Now the binomial
formula (1.6) shows that
If(x + h) - f(x)1 =
(4.5)
= Ix[n-l]h + x[n-2]h 2 /2! + ... + h n In!1
:S Ihl.{lxl[n-l] + Ixl[n-2]lhI/2! + ... + Ihl n - 1 In!}.
The expression between the braces { } differs from the expansion of (Ixl +
Ihl)[n-l] only in the presence, in the term in IhIP , of a denominator (p + I)!
instead of p!; since (p + I)! > p!, one concludes that
(4.6)
since Ihl is a factor of the right hand side, continuity is proved, since,
for Ihl < 1 for example, the right hand side is bounded by M/hl where
II - Convergence: Discrete variables
inequality III zl < r is satisfied as soon as Izl > l/r. Similarly, x 2 tends to 4
when x tends to 2, since on the one hand
Ix 2 - 41 = Ix - 21·lx + 21 < 51x - 21 for Ix - 21 ~ 1,
while on the other hand, 51x - 21 < r for Ix - 21 < r 15; so Ix 2 - 41 < r once
Ix - 21 < r' = min(1,rI5).
We shall hardly ever, in this chapter, use the concept of limit except for
the case of sequences, Le. of functions where the independent variable takes
only integer values. The general case which we have just mentioned will be
covered in detail in Chap. III. Since we shall nevertheless need to speak
occasionally of continuity and of differentiability in this chapter, let us now
give the definitions of these two fundamental properties.
A scalar (Le. complex-valued) function f defined on a set X C C is said
to be continuous at a point a of X if
limf(x) = f(a) when x ~ a.
This means that for all r > 0 there exists an r' > 0 such that
(4.4)
{(x E X) & (Ix - al < r'n =? If(x) - f(a)1 < r
or again that, for all r > 0, f(x) is constant to within r on a neighbourhood
of a in X or again, in decimal language, that if one wants to calculate f(a)
to within lO-n it suffices to calculate f(x) for any x E X having sufficiently
many decimal places in common with a, for example the number obtained
by replacing all the digits of a of sufficiently high rank by 0; this is what all
the practitioners of numerical analysis have always done, and this is what
all computers do. The calculation above, showing that x 2 tends to 4 when x
tends to 2, expresses the continuity of the function x r---+ x 2 at x = 2.
More generally, let us show, as a useful exercise, that the functions x n , or,
as clearly amounts to the same, f(x) = x[n] = xnln!, are continuous on C. It
suffices to show - the other formulation of continuity - that, for x given, the
difference If(x + h) - f(x)1 is < r for Ihl sufficiently small. Now the binomial
formula (1.6) shows that
If(x + h) - f(x)1 =
(4.5)
= Ix[n-l]h + x[n-2]h 2 /2! + ... + h n In!1
:S Ihl.{lxl[n-l] + Ixl[n-2]lhI/2! + ... + Ihl n - 1 In!}.
The expression between the braces { } differs from the expansion of (Ixl +
Ihl)[n-l] only in the presence, in the term in IhIP , of a denominator (p + I)!
instead of p!; since (p + I)! > p!, one concludes that
(4.6)
since Ihl is a factor of the right hand side, continuity is proved, since,
for Ihl < 1 for example, the right hand side is bounded by M/hl where